3.1 Graphing f (x) = ax2 3.2 Graphing f (x) = ax2 + c3.3 Graphing f (x) = ax2 + bx + c3.4 Graphing f (x) = a(x − h)2 + k3.5 Graphing f (x) = a(x − p)(x − q)3.6 Focus of a Parabola3.7 Comparing Linear, Exponential, and Quadratic Functions
3 Graphing Quadratic Functions
Roller Coaster (p. 152)
Firework Explosion (p. 141)
Garden Waterfalls (p. 134)
Breathing Rates (p. 177)
SEE the Big Idea
Electricity-Generating Dish (p. 165)
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121
Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyGraphing Linear Equations
Example 1 Graph y = −x − 1.
Step 1 Make a table of values.
x y = −x − 1 y (x, y)
−1 y = −(−1) − 1 0 (−1, 0)
0 y = −(0) − 1 −1 (0, −1)
1 y = −(1) − 1 −2 (1, −2)
2 y = −(2) − 1 −3 (2, −3)
Step 2 Plot the ordered pairs.
Step 3 Draw a line through the points.
Graph the linear equation.
1. y = 2x − 3 2. y = −3x + 4
3. y = − 1 — 2 x − 2 4. y = x + 5
Evaluating Expressions
Example 2 Evaluate 2x2 + 3x − 5 when x = −1.
2x2 + 3x − 5 = 2(−1)2 + 3(−1) − 5 Substitute −1 for x.
= 2(1) + 3(−1) − 5 Evaluate the power.
= 2 − 3 − 5 Multiply.
= −6 Subtract.
Evaluate the expression when x = −2.
5. 5x2 − 9 6. 3x2 + x − 2
7. −x2 + 4x + 1 8. x2 + 8x + 5
9. −2x2 − 4x + 3 10. −4x2 + 2x − 6
11. ABSTRACT REASONING Complete the table. Find a pattern in the differences of
consecutive y-values. Use the pattern to write an expression for y when x = 6.
x 1 2 3 4 5
y = ax2
x
y
2
−4
2
(−1, 0)(0, −1) (1, −2)
(2, −3)
x22
2
y = −x − 1
Dynamic Solutions available at BigIdeasMath.com
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122 Chapter 3 Graphing Quadratic Functions
Mathematical Mathematical PracticesPracticesProblem-Solving Strategies
Mathematically profi cient students try special cases of the original problem to gain insight into its solution.
Monitoring ProgressMonitoring ProgressGraph the quadratic function. Then describe its graph.
1. y = −x2 2. y = 2x2 3. f (x) = 2x2 + 1 4. f (x) = 2x2 − 1
5. f (x) = 1 —
2 x2 + 4x + 3 6. f (x) =
1 —
2 x2 − 4x + 3 7. y = −2(x + 1)2 + 1 8. y = −2(x − 1)2 + 1
9. How are the graphs in Monitoring Progress Questions 1−8 similar? How are they different?
Graphing the Parent Quadratic Function
Graph the parent quadratic function y = x2. Then describe its graph.
SOLUTIONThe function is of the form y = ax2, where a = 1. By plotting several points, you can see
that the graph is U-shaped, as shown.
2
4
6
8
x2 4 6−6 −4 −2
y
y = x2
The graph opens up, and the lowest point is at the origin.
Trying Special Cases When solving a problem in mathematics, it can be helpful to try special cases of the
original problem. For instance, in this chapter, you will learn to graph a quadratic
function of the form f (x) = ax2 + bx + c. The problem-solving strategy used is to
fi rst graph quadratic functions of the form f (x) = ax2. From there, you progress to
other forms of quadratic functions.
f (x) = ax2 Section 3.1
f (x) = ax2 + c Section 3.2
f (x) = ax2 + bx + c Section 3.3
f (x) = a(x − h)2 + k Section 3.4
f (x) = a(x − p)(x − q) Section 3.5
Core Core ConceptConcept
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Section 3.1 Graphing f(x) = ax2 123
Graphing f (x) = ax23.1
Essential QuestionEssential Question What are some of the characteristics of the
graph of a quadratic function of the form f (x) = ax2?
Graphing Quadratic Functions
Work with a partner. Graph each quadratic function. Compare each graph to the
graph of f (x) = x2.
a. g(x) = 3x2 b. g(x) = −5x2
2
4
6
8
10
x2 4 6−6 −4 −2
y
f(x) = x2
4
x2 4 6−6 −4 −2
−4
−8
−12
−16
y
f(x) = x2
c. g(x) = −0.2x2 d. g(x) = 1 —
10 x2
2
4
6
x2 4 6−6 −4 −2
−2
−4
−6
y
f(x) = x2
2
4
6
8
10
x2 4 6−6 −4 −2
y
f(x) = x2
Communicate Your AnswerCommunicate Your Answer 2. What are some of the characteristics of the graph of a quadratic function of
the form f (x) = ax2?
3. How does the value of a affect the graph of f (x) = ax2? Consider 0 < a < 1,
a > 1, −1 < a < 0, and a < −1. Use a graphing calculator to verify
your answers.
4. The fi gure shows the graph of a quadratic function
of the form y = ax2. Which of the intervals
in Question 3 describes the value of a? Explain
your reasoning.
REASONING QUANTITATIVELY
To be profi cient in math, you need to make sense of quantities and their relationships in problem situations.
6
−1
−6
7
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124 Chapter 3 Graphing Quadratic Functions
3.1 Lesson What You Will LearnWhat You Will Learn Identify characteristics of quadratic functions.
Graph and use quadratic functions of the form f (x) = ax2.
Identifying Characteristics of Quadratic FunctionsA quadratic function is a nonlinear function that can be written in the standard form
y = ax2 + bx + c, where a ≠ 0. The U-shaped graph of a quadratic function is called
a parabola. In this lesson, you will graph quadratic functions, where b and c equal 0.
Identifying Characteristics of a Quadratic Function
Consider the graph of the quadratic function.
Using the graph, you can identify characteristics such as the vertex, axis of symmetry,
and the behavior of the graph, as shown.
2
4
6
x42−4−6
y
axis ofsymmetry:x = −1
vertex:(−1, −2)
Function isdecreasing.
Function isincreasing.
You can also determine the following:
• The domain is all real numbers.
• The range is all real numbers greater than or equal to −2.
• When x < −1, y increases as x decreases.
• When x > −1, y increases as x increases.
quadratic function, p. 124parabola, p. 124vertex, p. 124axis of symmetry, p. 124
Previousdomainrangevertical shrinkvertical stretchrefl ection
Core VocabularyCore Vocabullarry
Core Core ConceptConceptCharacteristics of Quadratic FunctionsThe parent quadratic function is f (x) = x2. The graphs of all other quadratic
functions are transformations of the graph of the parent quadratic function.
The lowest point
on a parabola that
opens up or the
highest point on a
parabola that opens
down is the vertex.
The vertex of the
graph of f (x) = x2
is (0, 0).
x
y
axis ofsymmetry
vertex
decreasing increasing
The vertical line that
divides the parabola
into two symmetric
parts is the axis of symmetry. The axis
of symmetry passes
through the vertex. For
the graph of f (x) = x2,
the axis of symmetry
is the y-axis, or x = 0.
REMEMBERThe notation f (x) is another name for y.
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Section 3.1 Graphing f(x) = ax2 125
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Identify characteristics of the quadratic function and its graph.
1.
−2
−4
1
x642−2
y 2.
4
2
6
8
x1−3−7 −1
y
Graphing and Using f (x) = ax2
Core Core ConceptConceptGraphing f (x) = ax2 When a > 0• When 0 < a < 1, the graph of f (x) = ax2
is a vertical shrink of the graph of
f (x) = x2.
• When a > 1, the graph of f (x) = ax2 is a
vertical stretch of the graph of f (x) = x2.
Graphing f (x) = ax2 When a < 0• When −1 < a < 0, the graph of
f (x) = ax2 is a vertical shrink with a
refl ection in the x-axis of the graph
of f (x) = x2.
• When a < −1, the graph of f (x) = ax2
is a vertical stretch with a refl ection in
the x-axis of the graph of f (x) = x2.
REMEMBERThe graph of y = a ⋅ f (x) is a vertical stretch or shrink by a factor of a of the graph of y = f (x).
The graph of y = −f (x) is a refl ection in the x-axis of the graph of y = f (x).
Graphing y = ax2 When a > 0
Graph g(x) = 2x2. Compare the graph to the graph of f (x) = x2.
SOLUTION
Step 1 Make a table
of values.
Step 2 Plot the ordered pairs.
Step 3 Draw a smooth curve through the points.
Both graphs open up and have the same vertex, (0, 0), and the same axis of
symmetry, x = 0. The graph of g is narrower than the graph of f because the
graph of g is a vertical stretch by a factor of 2 of the graph of f.
x −2 −1 0 1 2
g(x) 8 2 0 2 8
x
a = 1 a > 1
0 < a < 1
y
x
a = −1 a < −1
−1 < a < 0
y
2
4
6
8
x42−2−4
y
g(x) = 2x2 f(x) = x2
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126 Chapter 3 Graphing Quadratic Functions
Graphing y = ax2 When a < 0
Graph h(x) = − 1 — 3 x2. Compare the graph to the graph of f (x) = x2.
SOLUTION
Step 1 Make a table of values.
x −6 −3 0 3 6
h(x) −12 −3 0 −3 −12
Step 2 Plot the ordered pairs.
Step 3 Draw a smooth curve through the points.
The graphs have the same vertex, (0, 0),
and the same axis of symmetry, x = 0, but
the graph of h opens down and is wider
than the graph of f. So, the graph of h is a
vertical shrink by a factor of 1 —
3 and a
refl ection in the x-axis of the graph of f.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Graph the function. Compare the graph to the graph of f (x) = x2.
3. g(x) = 5x2 4. h(x) = 1 —
3 x2 5. n(x) =
3 —
2 x2
6. p(x) = −3x2 7. q(x) = −0.1x2 8. g(x) = − 1 — 4 x2
Solving a Real-Life Problem
The diagram at the left shows the cross section of a satellite dish, where x and y are
measured in meters. Find the width and depth of the dish.
SOLUTION
Use the domain of the function to fi nd the width
of the dish. Use the range to fi nd the depth.
The leftmost point on the graph is (−2, 1), and
the rightmost point is (2, 1). So, the domain
is −2 ≤ x ≤ 2, which represents 4 meters.
The lowest point on the graph is (0, 0), and the highest points on the graph
are (−2, 1) and (2, 1). So, the range is 0 ≤ y ≤ 1, which represents 1 meter.
So, the satellite dish is 4 meters wide and 1 meter deep.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
9. The cross section of a spotlight can be modeled by the graph of y = 0.5x2,
where x and y are measured in inches and −2 ≤ x ≤ 2. Find the width and
depth of the spotlight.
STUDY TIPTo make the calculations easier, choose x-values that are multiples of 3.
−4
−8
−12
4
x84−4−8
y
f(x) = x2
h(x) = − x213
1
2
x2−1−2
y
y = x214(−2, 1)
(0, 0)
(2, 1)
width
depth
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Section 3.1 Graphing f(x) = ax2 127
Exercises3.1 Dynamic Solutions available at BigIdeasMath.com
In Exercises 3 and 4, identify characteristics of the quadratic function and its graph. (See Example 1.)
3.
−2
−6
1
x42−2
y
4.
4
8
12
x84−4−8
y
In Exercises 5–12, graph the function. Compare the graph to the graph of f (x) = x2. (See Examples 2 and 3.)
5. g(x) = 6x2 6. b(x) = 2.5x2
7. h(x) = 1 — 4 x2 8. j(x) = 0.75x2
9. m(x) = −2x2 10. q(x) = − 9 — 2 x2
11. k(x) = −0.2x2 12. p(x) = − 2 — 3 x2
In Exercises 13–16, use a graphing calculator to graph the function. Compare the graph to the graph of y = −4x2.
13. y = 4x2 14. y = −0.4x2
15. y = −0.04x2 16. y = −0.004x2
17. ERROR ANALYSIS Describe and correct the error in
graphing and comparing y = x2 and y = 0.5x2.
x
y = x2
y = 0.5x2
y
The graphs have the same vertex and the same
axis of symmetry. The graph of y = 0.5x2 is
narrower than the graph of y = x2.
✗
18. MODELING WITH MATHEMATICS The arch support of
a bridge can be modeled by y = −0.0012x2, where x
and y are measured in feet. Find the height and width
of the arch. (See Example 4.)
50
x350 45025050−50−250−350−450
y
−150
−250
−350
19. PROBLEM SOLVING The breaking strength z
(in pounds) of a manila rope can be modeled by
z = 8900d2, where d is the diameter (in inches)
of the rope.
a. Describe the domain and
range of the function.
b. Graph the function using
the domain in part (a).
c. A manila rope has four times
the breaking strength of another manila rope. Does
the stronger rope have four times the diameter?
Explain.
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. VOCABULARY What is the U-shaped graph of a quadratic function called?
2. WRITING When does the graph of a quadratic function open up? open down?
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
d
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128 Chapter 3 Graphing Quadratic Functions
20. HOW DO YOU SEE IT? Describe the possible values
of a.
a.
x
y
g(x) = ax2 f(x) = x2
b.
x
y
f(x) = x2
g(x) = ax2
ANALYZING GRAPHS In Exercises 21–23, use the graph.
x
f(x) = ax2, a > 0
g(x) = ax2, a < 0
y
21. When is each function increasing?
22. When is each function decreasing?
23. Which function could include the point (−2, 3)? Find
the value of a when the graph passes through (−2, 3).
24. REASONING Is the x-intercept of the graph of y = ax2
always 0? Justify your answer.
25. REASONING A parabola opens up and passes through
(−4, 2) and (6, −3). How do you know that (−4, 2) is
not the vertex?
ABSTRACT REASONING In Exercises 26–29, determine whether the statement is always, sometimes, or never true. Explain your reasoning.
26. The graph of f (x) = ax2 is narrower than the graph of
g(x) = x2 when a > 0.
27. The graph of f (x) = ax2 is narrower than the graph of
g(x) = x2 when ∣ a ∣ > 1.
28. The graph of f (x) = ax2 is wider than the graph of
g(x) = x2 when 0 < ∣ a ∣ < 1.
29. The graph of f (x) = ax2 is wider than the graph of
g(x) = dx2 when ∣ a ∣ > ∣ d ∣ .
30. THOUGHT PROVOKING Draw the isosceles triangle
shown. Divide each leg into eight congruent segments.
Connect the highest point of one leg with the lowest
point of the other leg. Then connect the second
highest point of one leg to the second lowest point of
the other leg. Continue this process. Write a quadratic
function whose graph models the shape that appears.
base
legleg
(−6, −4) (6, −4)
(0, 4)
31. MAKING AN ARGUMENT The diagram shows the
parabolic cross section
of a swirling glass of
water, where x and y are
measured in centimeters.
a. About how wide is the
mouth of the glass?
b. Your friend claims that
the rotational speed of
the water would have
to increase for the
cross section to be
modeled by y = 0.1x2.
Is your friend correct?
Explain your reasoning.
Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyEvaluate the expression when n = 3 and x = −2. (Skills Review Handbook)
32. n2 + 5 33. 3x2 − 9 34. −4n2 + 11 35. n + 2x2
Reviewing what you learned in previous grades and lessons
x
y(−4, 3.2) (4, 3.2)
(0, 0)
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Section 3.2 Graphing f(x) = ax2 + c 129
Graphing f (x) = ax2 + c3.2
Essential QuestionEssential Question How does the value of c affect the graph of
f (x) = ax2 + c?
Graphing y = ax2 + c
Work with a partner. Sketch the graphs of the functions in the same coordinate
plane. What do you notice?
a. f (x) = x2 and g(x) = x2 + 2 b. f (x) = 2x2 and g(x) = 2x2 – 2
2
4
6
8
10
x2 4 6−6 −4 −2
−2
y
2
4
6
8
10
x2 4 6−6 −4 −2
−2
y
Finding x-Intercepts of Graphs
Work with a partner. Graph each function. Find the x-intercepts of the graph.
Explain how you found the x-intercepts.
a. y = x2 − 7 b. y = −x2 + 1
2
4
x2 4 6−6 −4 −2
−2
−4
−6
−8
y
2
4
x2 4 6−6 −4 −2
−2
−4
−6
−8
y
Communicate Your AnswerCommunicate Your Answer 3. How does the value of c affect the graph of
f (x) = ax2 + c?
4. Use a graphing calculator to verify your answers
to Question 3.
5. The fi gure shows the graph of a quadratic function
of the form y = ax2 + c. Describe possible values
of a and c. Explain your reasoning.
USING TOOLS STRATEGICALLY
To be profi cient in math, you need to consider the available tools, such as a graphing calculator, when solving a mathematical problem.
6
−1
−6
7
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130 Chapter 3 Graphing Quadratic Functions
3.2 Lesson What You Will LearnWhat You Will Learn Graph quadratic functions of the form f (x) = ax2 + c.
Solve real-life problems involving functions of the form f (x) = ax2 + c.
Graphing f (x) = ax2 + c
Graphing y = x2 + c
Graph g(x) = x2 − 2. Compare the graph to the graph of f (x) = x2.
SOLUTION
Step 1 Make a table of values.
x −2 −1 0 1 2
g(x) 2 −1 −2 −1 2
Step 2 Plot the ordered pairs.
Step 3 Draw a smooth curve through the points.
−3
2
4
x42−2−4
y
f(x) = x2
−33
−22−44
g(x) = x2 − 2
Both graphs open up and have the same axis of symmetry, x = 0. The vertex
of the graph of g, (0, −2), is below the vertex of the graph of f, (0, 0), because the
graph of g is a vertical translation 2 units down of the graph of f.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Graph the function. Compare the graph to the graph of f (x) = x2.
1. g(x) = x2 − 5 2. h(x) = x2 + 3
REMEMBERThe graph of y = f (x) + k is a vertical translation, and the graph of y = f (x − h) is a horizontal translation of the graph of f.
zero of a function, p. 132
Previoustranslationvertex of a parabolaaxis of symmetryvertical stretchvertical shrink
Core VocabularyCore Vocabullarry
Core Core ConceptConceptGraphing f (x) = ax2 + c• When c > 0, the graph of f (x) = ax2 + c
is a vertical translation c units up of the
graph of f (x) = ax2.
• When c < 0, the graph of f (x) = ax2 + c
is a vertical translation ∣ c ∣ units down of
the graph of f (x) = ax2.
The vertex of the graph of f (x) = ax2 + c is
(0, c), and the axis of symmetry is x = 0.
x
c = 0c > 0
c < 0
y
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Section 3.2 Graphing f(x) = ax2 + c 131
Graphing y = ax2 + c
Graph g(x) = 4x2 + 1. Compare the graph to the graph of f (x) = x2.
SOLUTION
Step 1 Make a table of values.
x −2 −1 0 1 2
g(x) 17 5 1 5 17
Step 2 Plot the ordered pairs.
Step 3 Draw a smooth curve through the points.
Both graphs open up and have the same axis of symmetry, x = 0. The graph of
g is narrower, and its vertex, (0, 1), is above the vertex of the graph of f, (0, 0).
So, the graph of g is a vertical stretch by a factor of 4 and a vertical translation
1 unit up of the graph of f.
Translating the Graph of y = ax2 + c
Let f (x) = −0.5x2 + 2 and g(x) = f (x) − 7.
a. Describe the transformation from the graph of f to the graph of g. Then graph f and
g in the same coordinate plane.
b. Write an equation that represents g in terms of x.
SOLUTION
a. The function g is of the form y = f (x) + k, where k = −7. So, the graph of g is a
vertical translation 7 units down of the graph of f.
x −4 −2 0 2 4
f (x) −6 0 2 0 −6
g(x) −13 −7 −5 −7 −13
b. g(x) = f (x) − 7 Write the function g.
= −0.5x2 + 2 − 7 Substitute for f (x).
= −0.5x2 − 5 Subtract.
So, the equation g(x) = −0.5x2 − 5 represents g in terms of x.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Graph the function. Compare the graph to the graph of f (x) = x2.
3. g(x) = 2x2 − 5
4. h(x) = − 1 — 4 x2 + 4
5. Let f (x) = 3x2 − 1 and g(x) = f (x) + 3.
a. Describe the transformation from the graph of f to the graph of g. Then graph
f and g in the same coordinate plane.
b. Write an equation that represents g in terms of x.
−0.5x 2 + 2
f (x ) − 7
4
8
12
16
x42
y
f(x) = x2
g(x) = 4x2 + 1
−4
−8
−12
x84−4−8
y
fg
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132 Chapter 3 Graphing Quadratic Functions
Solving Real-Life ProblemsA zero of a function f is an x-value for which f (x) = 0. A zero of a function is an
x-intercept of the graph of the function.
Solving a Real-Life Problem
The function f(t) = 16t2 represents the distance (in feet) a dropped object falls in
t seconds. The function g(t) = s0 represents the initial height (in feet) of the object.
a. Find and interpret h(t) = −f(t) + g(t).
b. An egg is dropped from a height of 64 feet. After how many seconds does the egg
hit the ground?
c. Suppose the initial height is adjusted by k feet. How will this affect part (b)?
SOLUTION
a. h(t) = −f(t) + g(t) Write the function h.
= −(16t2) + s0 Substitute for f (t) and g(t).
= −16t2 + s0 Simplify.
The function h (t) = −16t2 + s0 represents the height (in feet) of a falling
object t seconds after it is dropped from an initial height s0 (in feet).
b. The initial height is 64 feet. So, the function h (t) = −16t2 + 64 represents the
height of the egg t seconds after it is dropped. The egg hits the ground when
h (t) = 0.
Step 1 Make a table of values and sketch the graph.
t 0 1 2
h (t) 64 48 0
Step 2 Find the positive zero of the function. When t = 2, h (t) = 0. So, the
zero is 2.
The egg hits the ground 2 seconds after it is dropped.
c. When the initial height is adjusted by k feet, the graph of h is translated up kunits when k > 0 or down ∣k ∣ units when k < 0. So, the x-intercept of the graph
of h will move right when k > 0 or left when k < 0.
When k > 0, the egg will take more than 2 seconds to hit the ground.
When k < 0, the egg will take less than 2 seconds to hit the ground.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
6. Explain why only nonnegative values of t are used in Example 4.
7. WHAT IF? The egg is dropped from a height of 100 feet. After how many seconds
does the egg hit the ground?
COMMON ERRORThe graph in Step 1 shows the height of the object over time, not the path of the object.
64 ft
16
32
48
t321
yh(t) = −16t2 + 64
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Section 3.2 Graphing f(x) = ax2 + c 133
Exercises3.2 Dynamic Solutions available at BigIdeasMath.com
In Exercises 3–6, graph the function. Compare the graph to the graph of f(x) = x2. (See Example 1.)
3. g(x) = x2 + 6 4. h(x) = x2 + 8
5. p(x) = x2 − 3 6. q(x) = x2 − 1
In Exercises 7–12, graph the function. Compare the graph to the graph of f(x) = x2. (See Example 2.)
7. g(x) = −x2 + 3 8. h(x) = −x2 − 7
9. s(x) = 2x2 − 4 10. t(x) = −3x2 + 1
11. p(x) = − 1 — 3 x2 − 2 12. q(x) =
1 —
2 x2 + 6
In Exercises 13–16, describe the transformation from the graph of f to the graph of g. Then graph f and g in the same coordinate plane. Write an equation that represents g in terms of x. (See Example 3.)
13. f (x) = 3x2 + 4 14. f (x) = 1 —
2 x2 + 1
g(x) = f (x) + 2 g(x) = f (x) − 4
15. f (x) = − 1 — 4 x2 − 6 16. f (x) = 4x2 − 5
g(x) = f (x) − 3 g(x) = f (x) + 7
17. ERROR ANALYSIS Describe and correct the error in
comparing the graphs.
4
6
x2−2
y
y = x2y = 3x2 + 2
The graph of y = 3x2 + 2 is a vertical
shrink by a factor of 3 and a translation
2 units up of the graph of y = x2.
✗
18. ERROR ANALYSIS Describe and correct the error in
graphing and comparing f (x) = x2 and g(x) = x2 − 10.
20
−10
x4−4
y
fg
Both graphs open up and have the same axis
of symmetry. However, the vertex of the graph
of g, (0, 10), is 10 units above the vertex of
the graph of f, (0, 0).
✗
In Exercises 19–26, fi nd the zeros of the function.
19. y = x2 − 1 20. y = x2 − 36
21. f (x) = −x2 + 25 22. f (x) = −x2 + 49
23. f (x) = 4x2 − 16 24. f (x) = 3x2 − 27
25. f (x) = −12x2 + 3 26. f (x) = −8x2 + 98
27. MODELING WITH MATHEMATICS A water balloon is
dropped from a height of 144 feet. (See Example 4.)
a. What does the function h(t) = −16t2 + 144
represent in this situation?
b. After how many seconds does the water balloon
hit the ground?
c. Suppose the initial height is adjusted by k feet.
How does this affect part (b)?
28. MODELING WITH MATHEMATICS The function
y = −16x2 + 36 represents the height y (in feet) of
an apple x seconds after falling from a tree. Find and
interpret the x- and y-intercepts.
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. VOCABULARY State the vertex and axis of symmetry of the graph of y = ax2 + c.
2. WRITING How does the graph of y = ax2 + c compare to the graph of y = ax2?
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
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134 Chapter 3 Graphing Quadratic Functions
x
y
(0, 4)
(2, 0)(−2, 0)
In Exercises 29–32, sketch a parabola with the given characteristics.
29. The parabola opens up, and the vertex is (0, 3).
30. The vertex is (0, 4), and one of the x-intercepts is 2.
31. The related function is increasing when x < 0, and the
zeros are −1 and 1.
32. The highest point on the parabola is (0, −5).
33. DRAWING CONCLUSIONS You and your friend
both drop a ball at the same time. The function
h(x) = −16x2 + 256 represents the height (in feet)
of your ball after x seconds. The function
g(x) = −16x2 + 300 represents the height (in feet)
of your friend’s ball after x seconds.
a. Write the function T(x) = h(x) − g(x). What does
T(x) represent?
b. When your ball hits the ground, what is the
height of your friend’s ball? Use a graph to justify
your answer.
34. MAKING AN ARGUMENT Your friend claims that in
the equation y = ax2 + c, the vertex changes when
the value of a changes. Is your friend correct? Explain
your reasoning.
35. MATHEMATICAL CONNECTIONS The area A
(in square feet) of a square patio is represented by
A = x2, where x is the length of one side of the patio.
You add 48 square feet to the patio, resulting in a total
area of 192 square feet. What are the dimensions of
the original patio? Use a graph to justify your answer.
36. HOW DO YOU SEE IT? The graph of f (x) = ax2 + c
is shown. Points A and B are the same distance from
the vertex of the graph of f. Which point is closer to
the vertex of the graph of f as c increases?
x
y
fA
B
37. REASONING Describe two methods you can use to
fi nd the zeros of the function f (t) = −16t2 + 400.
Check your answer by graphing.
38. PROBLEM SOLVING The paths of water from three
different garden waterfalls are given below. Each
function gives the height h (in feet) and the horizontal
distance d (in feet) of the water.
Waterfall 1 h = −3.1d 2 + 4.8
Waterfall 2 h = −3.5d 2 + 1.9
Waterfall 3 h = −1.1d 2 + 1.6
a. Which waterfall drops water
from the highest point?
b. Which waterfall follows the
narrowest path?
c. Which waterfall sends water the farthest?
39. WRITING EQUATIONS Two acorns fall to the ground
from an oak tree. One falls 45 feet, while the other
falls 32 feet.
a. For each acorn, write an equation that represents
the height h (in feet) as a function of the time t (in seconds).
b. Describe how the graphs of the two equations
are related.
40. THOUGHT PROVOKING One of two
classic problems in calculus is
to fi nd the area under a curve.
Approximate the area of the
region bounded by the
parabola and the x-axis.
Show your work.
41. CRITICAL THINKING A cross section of the
parabolic surface of
the antenna shown
can be modeled by
y = 0.012x2, where
x and y are measured
in feet. The antenna is moved up so that the outer
edges of the dish are 25 feet above the x-axis. Where
is the vertex of the cross section located? Explain.
Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyEvaluate the expression when a = 4 and b = −3. (Skills Review Handbook)
42. a — 4b
43. − b —
2a 44.
a − b — 3a + b
45. − b + 2a
— ab
Reviewing what you learned in previous grades and lessons
20−20
10
30
−40 40 x
y
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Section 3.3 Graphing f(x) = ax2 + bx + c 135
Graphing f (x) = ax2 + bx + c3.3
Essential QuestionEssential Question How can you fi nd the vertex of the graph
of f (x) = ax2 + bx + c?
Comparing x-Intercepts with the Vertex
Work with a partner.
a. Sketch the graphs of y = 2x2 − 8x and y = 2x2 − 8x + 6.
b. What do you notice about the x-coordinate of the vertex of each graph?
c. Use the graph of y = 2x2 − 8x to fi nd its x-intercepts. Verify your answer
by solving 0 = 2x2 − 8x.
d. Compare the value of the x-coordinate of the vertex with the values of
the x-intercepts.
Finding x-Intercepts
Work with a partner.
a. Solve 0 = ax2 + bx for x by factoring.
b. What are the x-intercepts of the graph of y = ax2 + bx?
c. Copy and complete the table to verify your answer.
x y = ax2 + bx
0
− b —
a
Deductive Reasoning
Work with a partner. Complete the following logical argument.
The x-intercepts of the graph of y = ax2 + bx are 0 and − b —
a .
The vertex of the graph of y = ax2 + bx occurs when x = .
The vertex of the graph of y = ax2 + bx + c occurs when x = .
The vertices of the graphs of y = ax2 + bx and y = ax2 + bx + c
have the same x-coordinate.
Communicate Your AnswerCommunicate Your Answer 4. How can you fi nd the vertex of the graph of f (x) = ax2 + bx + c?
5. Without graphing, fi nd the vertex of the graph of f (x) = x2 − 4x + 3. Check your
result by graphing.
CONSTRUCTINGVIABLEARGUMENTS
To be profi cient in math, you need to make conjectures and build a logical progression of statements.
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136 Chapter 3 Graphing Quadratic Functions
3.3 Lesson What You Will LearnWhat You Will Learn Graph quadratic functions of the form f (x) = ax2 + bx + c.
Find maximum and minimum values of quadratic functions.
Graphing f (x) = ax2 + bx + c
Finding the Axis of Symmetry and the Vertex
Find (a) the axis of symmetry and (b) the vertex of the graph of f (x) = 2x2 + 8x − 1.
SOLUTION
a. Find the axis of symmetry when a = 2 and b = 8.
x = − b —
2a Write the equation for the axis of symmetry.
x = − 8 —
2(2) Substitute 2 for a and 8 for b.
x = −2 Simplify.
The axis of symmetry is x = −2.
b. The axis of symmetry is x = −2, so the x-coordinate of the vertex is −2. Use the
function to fi nd the y-coordinate of the vertex.
f (x) = 2x2 + 8x − 1 Write the function.
f (−2) = 2(−2)2 + 8(−2) − 1 Substitute −2 for x.
= −9 Simplify.
The vertex is (−2, −9).
Monitoring Progress Help in English and Spanish at BigIdeasMath.com
Find (a) the axis of symmetry and (b) the vertex of the graph of the function.
1. f (x) = 3x2 − 2x 2. g(x) = x2 + 6x + 5 3. h(x) = − 1 — 2 x2 + 7x − 4
Check
2
−12
−6
4
X=-2 Y=-9
6
4f(x) = 2x2 + 8x − 1
maximum value, p. 137minimum value, p. 137
Previousindependent variabledependent variable
Core VocabularyCore Vocabullarry
Core Core ConceptConceptGraphing f (x) = ax2 + bx + c• The graph opens up when a > 0,
and the graph opens down
when a < 0.
• The y-intercept is c.
• The x-coordinate of
the vertex is − b —
2a .
• The axis of symmetry is
x = − b —
2a .
x
y
vertex
(0, c)
f(x) = ax2 + bx + c,where a > 0
x = − b2a
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Section 3.3 Graphing f(x) = ax2 + bx + c 137
Graphing f (x) = ax2 + bx + c
Graph f (x) = 3x2 − 6x + 5. Describe the domain and range.
SOLUTION
Step 1 Find and graph the axis of symmetry.
x = − b —
2a = −
(−6) —
2(3) = 1 Substitute and simplify.
Step 2 Find and plot the vertex.
The axis of symmetry is x = 1, so the x-coordinate of the vertex is 1. Use the
function to fi nd the y-coordinate of the vertex.
f (x) = 3x2 − 6x + 5 Write the function.
f (1) = 3(1)2 − 6(1) + 5 Substitute 1 for x.
= 2 Simplify.
So, the vertex is (1, 2).
Step 3 Use the y-intercept to fi nd two more points on the graph.
Because c = 5, the y-intercept is 5. So, (0, 5) lies on the graph. Because the
axis of symmetry is x = 1, the point (2, 5) also lies on the graph.
Step 4 Draw a smooth curve through the points.
The domain is all real numbers. The range is y ≥ 2.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Graph the function. Describe the domain and range.
4. h(x) = 2x2 + 4x + 1 5. k(x) = x2 − 8x + 7 6. p(x) = −5x2 − 10x − 2
Finding Maximum and Minimum Values
COMMON ERRORBe sure to include the negative sign before the fraction when fi nding the axis of symmetry.
REMEMBERThe domain is the set of all possible input values of the independent variable x. The range is the set of all possible output values of the dependent variable y.
Core Core ConceptConceptMaximum and Minimum ValuesThe y-coordinate of the vertex of the graph of f (x) = ax2 + bx + c is the
maximum value of the function when a < 0 or the minimum value of the
function when a > 0.
f (x) = ax2 + bx + c, a < 0 f (x) = ax2 + bx + c, a > 0
x
y maximum
x
y
minimum
2
4
x4 62
y
f(x) = 3x2 − 6x + 5
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138 Chapter 3 Graphing Quadratic Functions
Finding a Maximum or Minimum Value
Tell whether the function f (x) = −4x2 − 24x − 19 has a minimum value or a
maximum value. Then fi nd the value.
SOLUTION
For f (x) = −4x2 − 24x − 19, a = −4 and −4 < 0. So, the parabola opens down and
the function has a maximum value. To fi nd the maximum value, fi nd the y-coordinate
of the vertex.
First, fi nd the x-coordinate of the vertex. Use a = −4 and b = −24.
x = − b —
2a = −
(−24) —
2(−4) = −3 Substitute and simplify.
Then evaluate the function when x = −3 to fi nd the y-coordinate of the vertex.
f (−3) = −4(−3)2 − 24(−3) − 19 Substitute −3 for x.
= 17 Simplify.
The maximum value is 17.
Finding a Minimum Value
The suspension cables between the two towers of the Mackinac Bridge in Michigan
form a parabola that can be modeled by y = 0.000098x2 − 0.37x + 552, where x and y
are measured in feet. What is the height of the cable above the water at its lowest point?
500 1000 1500 2000 2500 3000 3500 x
−500
500
y
SOLUTION
The lowest point of the cable is at the vertex of the parabola. Find the x-coordinate of
the vertex. Use a = 0.000098 and b = −0.37.
x = − b —
2a = −
(−0.37) —
2(0.000098) ≈ 1888 Substitute and use a calculator.
Substitute 1888 for x in the equation to fi nd the y-coordinate of the vertex.
y = 0.000098(1888)2 − 0.37(1888) + 552 ≈ 203
The cable is about 203 feet above the water at its lowest point.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Tell whether the function has a minimum value or a maximum value. Then fi nd the value.
7. g(x) = 8x2 − 8x + 6 8. h(x) = − 1 — 4 x2 + 3x + 1
9. The cables between the two towers of the Tacoma Narrows Bridge in Washington
form a parabola that can be modeled by y = 0.00016x2 − 0.46x + 507, where
x and y are measured in feet. What is the height of the cable above the water at
its lowest point?
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Section 3.3 Graphing f(x) = ax2 + bx + c 139
Modeling with Mathematics
A group of friends is launching water balloons.
The function f (t) = −16t 2 + 80t + 5 represents the
height (in feet) of the fi rst water balloon t seconds after
it is launched. The height of the second water balloon
t seconds after it is launched is shown in the graph.
Which water balloon went higher?
SOLUTION
1. Understand the Problem You are given a function that represents the height
of the fi rst water balloon. The height of the second water balloon is represented
graphically. You need to fi nd and compare the maximum heights of the
water balloons.
2. Make a Plan To compare the maximum heights, represent both functions
graphically. Use a graphing calculator to graph f (t) = −16t2 + 80t + 5 in
an appropriate viewing window. Then visually compare the heights of the
water balloons.
3. Solve the Problem Enter the function f (t) = −16t2 + 80t + 5 into your
calculator and graph it. Compare the graphs to determine which function has
a greater maximum value.
00
150
6
1st water balloon
0 2 4 6 t
y
0
50
100
2nd water balloon
You can see that the second water balloon reaches a height of about 125 feet,
while the fi rst water balloon reaches a height of only about 100 feet.
So, the second water balloon went higher.
4. Look Back Use the maximum feature to determine that the maximum value of
f (t) = −16t2 + 80t + 5 is 105. Use a straightedge to represent a height of 105 feet
on the graph that represents the second water balloon to clearly see that the second
water balloon went higher.
00
150
6MaximumX=2.4999988 Y=105
1st water balloon
0 2 4 6 t
y
0
50
100
2nd water balloon
100
in.123456
cm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
10. Which balloon is in the air longer? Explain your reasoning.
11. Which balloon reaches its maximum height faster? Explain your reasoning.
MODELING WITH MATHEMATICS
Because time cannot be negative, use only nonnegative values of t.
ODELING WITH
Whic
SOL
1. Unof
gr
waWITHMOODELING WG W
0 2 4 6 t
y
0
50
100
2nd balloon
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140 Chapter 3 Graphing Quadratic Functions
Dynamic Solutions available at BigIdeasMath.com
1. VOCABULARY Explain how you can tell whether a quadratic function has a maximum value or a minimum value
without graphing the function.
2. DIFFERENT WORDS, SAME QUESTION Consider the quadratic function f (x) = −2x2 + 8x + 24. Which is
different? Find “both” answers.
What is the maximum value of the function?
What is the y-coordinate of the vertex of the graph of the function?
What is the greatest number in the range of the function?
What is the axis of symmetry of the graph of the function?
Exercises3.3
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
In Exercises 3–6, fi nd the vertex, the axis of symmetry, and the y-intercept of the graph.
3.
1
4
x2 4
y 4.
2
−2
−2−4 x
y
5. 1
−4
−4 1 x
y 6.
1
5
x2−2−4
y
In Exercises 7–12, fi nd (a) the axis of symmetry and (b) the vertex of the graph of the function. (See Example 1.)
7. f (x) = 2x2 − 4x 8. y = 3x2 + 2x
9. y = −9x2 − 18x − 1 10. f (x) = −6x2 + 24x − 20
11. f (x) = 2 —
5 x2 − 4x + 14 12. y = − 3 —
4 x2 + 9x − 18
In Exercises 13–18, graph the function. Describe the domain and range. (See Example 2.)
13. f (x) = 2x2 + 12x + 4 14. y = 4x2 + 24x + 13
15. y = −8x2 − 16x − 9 16. f (x) = −5x2 + 20x − 7
17. y = 2 —
3 x2 − 6x + 5 18. f (x) = − 1 —
2 x2 − 3x − 4
19. ERROR ANALYSIS Describe and correct the error
in fi nding the axis of symmetry of the graph of
y = 3x2 − 12x + 11.
x = − b —
2a =
−12 —
2(3) = −2
The axis of symmetry is x = −2.
✗
20. ERROR ANALYSIS Describe and correct the error in
graphing the function f (x) = x2 + 4x + 3.
The axis of symmetry
is x = b —
2a =
4 —
2(1) = 2.
f (2) = 22 + 4(2) + 3 = 15
So, the vertex is (2, 15).
The y-intercept is 3. So, the
points (0, 3) and (4, 3) lie on the graph.
✗6
12
x2 4
y
In Exercises 21–26, tell whether the function has a minimum value or a maximum value. Then fi nd the value. (See Example 3.)
21. y = 3x2 − 18x + 15 22. f (x) = −5x2 + 10x + 7
23. f (x) = −4x2 + 4x − 2 24. y = 2x2 − 10x + 13
25. y = − 1 — 2 x2 − 11x + 6 26. f (x) =
1 —
5 x2 − 5x + 27
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
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Section 3.3 Graphing f(x) = ax2 + bx + c 141
27. MODELING WITH MATHEMATICS The function shown
represents the height h (in feet) of a fi rework t seconds
after it is launched. The fi rework explodes at its
highest point. (See Example 4.)
h = −16t2 + 128t
a. When does the fi rework explode?
b. At what height does the fi rework explode?
28. MODELING WITH MATHEMATICS The function
h(t) = −16t 2 + 16t represents the height (in feet) of a
horse t seconds after it jumps during a steeplechase.
a. When does the horse reach its maximum height?
b. Can the horse clear a fence that is 3.5 feet tall?
If so, by how much?
c. How long is the horse in the air?
29. MODELING WITH MATHEMATICS The cable between
two towers of a suspension bridge can be modeled
by the function shown, where x and y are measured
in feet. The cable is at road level midway between
the towers.
x
y
y = x2 − x + 1501400
a. How far from each tower shown is the lowest point
of the cable?
b. How high is the road above the water?
c. Describe the domain and range of the function
shown.
30. REASONING Find the axis of symmetry of the graph
of the equation y = ax2 + bx + c when b = 0. Can
you fi nd the axis of symmetry when a = 0? Explain.
31. ATTENDING TO PRECISION The vertex of a parabola
is (3, −1). One point on the parabola is (6, 8). Find
another point on the parabola. Justify your answer.
32. MAKING AN ARGUMENT Your friend claims that it
is possible to draw a parabola through any two points
with different x-coordinates. Is your friend correct?
Explain.
USING TOOLS In Exercises 33–36, use the minimumor maximum feature of a graphing calculator to approximate the vertex of the graph of the function.
33. y = 0.5x2 + √—2 x − 3
34. y = −6.2x2 + 4.8x − 1
35. y = −πx2 + 3x
36. y = 0.25x2 − 52/3x + 2
37. MODELING WITH MATHEMATICS The opening of one
aircraft hangar is a parabolic arch that can be modeled
by the equation y = −0.006x2 + 1.5x, where x and yare measured in feet. The opening of a second aircraft
hangar is shown in the graph. (See Example 5.)
50 100 150 200 x
25
100y
a. Which aircraft hangar is taller?
b. Which aircraft hangar is wider?
38. MODELING WITH MATHEMATICS An offi ce
supply store sells about 80 graphing calculators
per month for $120 each. For each $6 decrease in
price, the store expects to sell eight more calculators.
The revenue from calculator sales is given by
the function R(n) = (unit price)(units sold), or
R(n) = (120 − 6n)(80 + 8n), where n is the
number of $6 price decreases.
a. How much should the
store charge to maximize
monthly revenue?
b. Using a different revenue
model, the store expects to
sell fi ve more calculators
for each $4 decrease in
price. Which revenue
model results in a greater
maximum monthly
revenue? Explain.
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142 Chapter 3 Graphing Quadratic Functions
MATHEMATICAL CONNECTIONS In Exercises 39 and 40, (a) fi nd the value of x that maximizes the area of the fi gure and (b) fi nd the maximum area.
39.
x in.
x in.6 in.
1.5x in.
40.
(x + 2) ft
12 ft
(12 − 4x) ft
41. WRITING Compare the graph of g(x) = x2 + 4x + 1
with the graph of h(x) = x2 − 4x + 1.
42. HOW DO YOU SEE IT? During an archery
competition, an archer shoots an arrow. The arrow
follows the parabolic path shown, where x and y are
measured in meters.
20 40 60 80 100 x
y
1
2
a. What is the initial height of the arrow?
b. Estimate the maximum height of the arrow.
c. How far does the arrow travel?
43. USING TOOLS The graph of a quadratic function
passes through (3, 2), (4, 7), and (9, 2). Does the
graph open up or down? Explain your reasoning.
44. REASONING For a quadratic function f, what does
f ( − b —
2a ) represent? Explain your reasoning.
45. PROBLEM SOLVING Write a function of the form
y = ax2 + bx whose graph contains the points (1, 6)
and (3, 6).
46. CRITICAL THINKING Parabolas A and B contain
the points shown. Identify characteristics of each
parabola, if possible. Explain your reasoning.
Parabola A
x y
2 3
6 4
Parabola B
x y
1 4
3 −4
5 4
47. MODELING WITH MATHEMATICS At a basketball
game, an air cannon launches T-shirts into the crowd.
The function y = − 1 —
8 x2 + 4x represents the path of
a T-shirt. The function 3y = 2x − 14 represents the
height of the bleachers. In both functions, y represents
vertical height (in feet) and x represents horizontal
distance (in feet). At what height does the T-shirt land
in the bleachers?
48. THOUGHT PROVOKING
One of two classic
problems in calculus is
fi nding the slope of a
tangent line to a curve.
An example of a tangent
line, which just touches
the parabola at one point, is shown.
Approximate the slope of the tangent line to the graph
of y = x2 at the point (1, 1). Explain your reasoning.
49. PROBLEM SOLVING The owners of a dog shelter want
to enclose a rectangular play area on the side of their
building. They have k feet of fencing. What is the
maximum area of the outside enclosure in terms of k?
(Hint: Find the y-coordinate of the vertex of the graph
of the area function.)
dog shelterbuilding
playarea
ww
Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyDescribe the transformation(s) from the graph of f (x) = ∣ x ∣ to the graph of the given function. (Section 1.1)
50. q(x) = ∣ x + 6 ∣ 51. h(x) = −0.5 ∣ x ∣ 52. g(x) = ∣ x − 2 ∣ + 5 53. p(x) = 3 ∣ x + 1 ∣
Reviewing what you learned in previous grades and lessons
−4 −2 x
2
ytangent line
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143
3.1–3.3 What Did You Learn?
Core VocabularyCore Vocabularyquadratic function, p. 124parabola, p. 124vertex, p. 124
axis of symmetry, p. 124zero of a function, p. 132
maximum value, p. 137minimum value, p. 137
Core ConceptsCore ConceptsSection 3.1Characteristics of Quadratic Functions, p. 124Graphing f (x) = ax2 When a > 0, p. 125Graphing f (x) = ax2 When a < 0, p. 125
Section 3.2Graphing f (x) = ax2 + c, p. 130
Section 3.3Graphing f (x) = ax2 + bx + c, p. 136Maximum and Minimum Values, p. 137
Mathematical PracticesMathematical Practices1. Explain your plan for solving Exercise 18 on page 127.
2. How does graphing the function in Exercise 27 on page 133 help you answer
the questions?
3. What defi nition and characteristics of the graph of a quadratic function did you
use to answer Exercise 44 on page 142?
• Draw a picture of a word problem before writing a verbal model. You do not have to be an artist.
• When making a review card for a word problem, include a picture. This will help you recall the information while taking a test.
• Make sure your notes are visually neat for easy recall.
Learning Visually
1434333333
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144 Chapter 3 Graphing Quadratic Functions
3.1–3.3 Quiz
Identify characteristics of the quadratic function and its graph. (Section 3.1)
1.
−2
−4
4
x42−2−4
y 2.
4
12
16
x84−4−8
y
Graph the function. Compare the graph to the graph of f (x) = x2. (Section 3.1 and Section 3.2)
3. h(x) = −x2 4. p(x) = 2x2 + 2
5. r(x) = 4x2 − 16 6. b(x) = 8x2
7. g(x) = 2 —
5 x2 8. m(x) = − 1 —
2 x2 − 4
Describe the transformation from the graph of f to the graph of g. Then graph f and g in the same coordinate plane. Write an equation that represents g in terms of x. (Section 3.2)
9. f (x) = 2x2 + 1; g(x) = f (x) + 2 10. f (x) = −3x2 + 12; g(x) = f (x) − 9
11. f (x) = 1 —
2 x2 − 2; g(x) = f (x) − 6 12. f (x) = 5x2 − 3; g(x) = f (x) + 1
Graph the function. Describe the domain and range. (Section 3.3)
13. f (x) = −4x2 − 4x + 7 14. f (x) = 2x2 + 12x + 5
15. y = x2 + 4x − 5 16. y = −3x2 + 6x + 9
Tell whether the function has a minimum value or a maximum value. Then fi nd the value. (Section 3.3)
17. f (x) = 5x2 + 10x − 3 18. f (x) = − 1 — 2 x2 + 2x + 16
19. y = −x2 + 4x + 12 20. y = 2x2 + 8x + 3
21. The distance y (in feet) that a coconut falls after t seconds is given by the function y = 16t 2. Use
a graph to determine how many seconds it takes for the coconut to fall 64 feet. (Section 3.1)
22. The function y = −16t 2 + 25 represents the height y (in feet) of a pinecone
t seconds after falling from a tree. (Section 3.2)
a. After how many seconds does the pinecone hit the ground?
b. A second pinecone falls from a height of 36 feet. Which pinecone hits
the ground in the least amount of time? Explain.
23. The function shown models the height (in feet) of a softball t seconds after it
is pitched in an underhand motion. Describe the domain and range. Find the
maximum height of the softball. (Section 3.3) t
hh(t) = −16t2 + 32t + 2
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Section 3.4 Graphing f (x) = a(x − h)2 + k 145
Graphing f (x) = a(x − h)2 + k3.4
Essential QuestionEssential Question How can you describe the graph of
f (x) = a(x − h)2?
Graphing y = a(x − h)2 When h > 0
Work with a partner. Sketch the graphs of the functions in the same coordinate
plane. How does the value of h affect the graph of y = a(x − h)2?
a. f (x) = x2 and g(x) = (x − 2)2 b. f (x) = 2x2 and g(x) = 2(x − 2)2
2
4
6
8
10
x2 4 6−6 −4 −2
−2
y
2
4
6
8
10
x2 4 6−6 −4 −2
−2
y
Graphing y = a(x − h)2 When h < 0
Work with a partner. Sketch the graphs of the functions in the same coordinate
plane. How does the value of h affect the graph of y = a(x − h)2?
a. f (x) = −x2 and g(x) = −(x + 2)2 b. f (x) = −2x2 and g(x) = −2(x + 2)2
2
4
x2 4 6−6 −4 −2
−2
−4
−6
−8
y
2
4
x2 4 6−6 −4 −2
−2
−4
−6
−8
y
Communicate Your AnswerCommunicate Your Answer 3. How can you describe the graph of f (x) = a(x − h)2?
4. Without graphing, describe the graph of each function. Use a graphing calculator
to check your answer.
a. y = (x − 3)2
b. y = (x + 3)2
c. y = −(x − 3)2
USING TOOLS STRATEGICALLY
To be profi cient in math, you need to consider the available tools, such as a graphing calculator, when solving a mathematical problem.
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146 Chapter 3 Graphing Quadratic Functions
3.4 Lesson What You Will LearnWhat You Will Learn Identify even and odd functions.
Graph quadratic functions of the form f (x) = a(x − h)2.
Graph quadratic functions of the form f (x) = a(x − h)2 + k.
Model real-life problems using f (x) = a(x − h)2 + k.
Identifying Even and Odd Functions
Identifying Even and Odd Functions
Determine whether each function is even, odd, or neither.
a. f (x) = 2x b. g(x) = x2 − 2 c. h(x) = 2x2 + x − 2
SOLUTION
a. f (x) = 2x Write the original function.
f (−x) = 2(−x) Substitute −x for x.
= −2x Simplify.
= −f (x) Substitute f (x) for 2x.
Because f (−x) = −f (x), the function is odd.
b. g(x) = x2 − 2 Write the original function.
g(−x) = (−x)2 − 2 Substitute −x for x.
= x2 − 2 Simplify.
= g(x) Substitute g(x) for x2 − 2.
Because g(−x) = g(x), the function is even.
c. h(x) = 2x2 + x − 2 Write the original function.
h(−x) = 2(−x)2 + (−x) − 2 Substitute −x for x.
= 2x2 − x − 2 Simplify.
Because h(x) = 2x2 + x − 2 and −h(x) = −2x2 − x + 2, you can conclude
that h(−x) ≠ h(x) and h(−x) ≠ −h(x). So, the function is neither even nor odd.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Determine whether the function is even, odd, or neither.
1. f (x) = 5x 2. g(x) = 2x 3. h(x) = 2x2 + 3
STUDY TIPMost functions are neither even nor odd.
even function, p. 146odd function, p. 146vertex form (of a quadratic
function), p. 148
Previousrefl ection
Core VocabularyCore Vocabullarry
Core Core ConceptConceptEven and Odd FunctionsA function y = f (x) is even when f (−x) = f (x) for each x in the domain of f. The graph of an even function is symmetric about the y-axis.
A function y = f (x) is odd when f (−x) = −f (x) for each x in the domain of f. The graph of an odd function is symmetric about the origin. A graph is symmetric about the origin when it looks the same after refl ections in the x-axis and then in
the y-axis.
STUDY TIPThe graph of an odd function looks the same after a 180° rotation about the origin.
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Section 3.4 Graphing f (x) = a(x − h)2 + k 147
Graphing y = a(x − h)2
Graph g(x) = 1 —
2 (x − 4)2. Compare the graph to the graph of f (x) = x2.
SOLUTION
Step 1 Graph the axis of symmetry. Because h = 4, graph x = 4.
Step 2 Plot the vertex. Because h = 4, plot (4, 0).
Step 3 Find and plot two more points on the graph. Choose two x-values less than
the x-coordinate of the vertex. Then fi nd g(x) for each x-value.
When x = 0: When x = 2:
g(0) = 1 —
2 (0 − 4)2 g(2) =
1 —
2 (2 − 4)2
= 8 = 2
So, plot (0, 8) and (2, 2).
Step 4 Refl ect the points plotted in Step 3 in the axis of symmetry. So, plot (8, 8)
and (6, 2).
Step 5 Draw a smooth curve
through the points.
Both graphs open up. The graph of g is wider than the graph of f. The axis of
symmetry x = 4 and the vertex (4, 0) of the graph of g are 4 units right of the axis
of symmetry x = 0 and the vertex (0, 0) of the graph of f. So, the graph of g is a
translation 4 units right and a vertical shrink by a factor of 1 —
2 of the graph of f.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Graph the function. Compare the graph to the graph of f (x) = x2.
4. g(x) = 2(x + 5)2 5. h(x) = −(x − 2)2
ANOTHER WAYIn Step 3, you could instead choose two x-values greater than the x-coordinate of the vertex.
STUDY TIPFrom the graph, you can see that f (x) = x2 is an even function. However, g(x) = 1 — 2 (x − 4)2 is neither even nor odd.
Graphing f (x) = a(x − h)2
Core Core ConceptConceptGraphing f (x) = a(x − h)2
• When h > 0, the graph of f (x) = a(x − h)2
is a horizontal translation h units right of the
graph of f (x) = ax2.
• When h < 0, the graph of f (x) = a(x − h)2
is a horizontal translation ∣ h ∣ units left of the
graph of f (x) = ax2.
The vertex of the graph of f (x) = a(x − h)2 is
(h, 0), and the axis of symmetry is x = h.
x
h = 0
h > 0h < 0
y
8
4
x8 12−4
y
f(x) = x2 g(x) = (x − 4)212
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148 Chapter 3 Graphing Quadratic Functions
Graphing y = a(x − h)2 + k
Graph g(x) = −2(x + 2)2 + 3. Compare the graph to the graph of f (x) = x2.
SOLUTION
Step 1 Graph the axis of symmetry. Because h = −2, graph x = −2.
Step 2 Plot the vertex. Because h = −2 and k = 3, plot (−2, 3).
Step 3 Find and plot two more points on the graph.
Choose two x-values less than the x-coordinate
of the vertex. Then fi nd g(x) for each x-value.
So, plot (−4, −5) and (−3, 1).
Step 4 Refl ect the points plotted in Step 3 in the axis of symmetry. So, plot (−1, 1)
and (0, −5).
Step 5 Draw a smooth curve through the points.
The graph of g opens down and is narrower than the graph of f. The vertex of the
graph of g, (−2, 3), is 2 units left and 3 units up of the vertex of the graph of f , (0, 0). So, the graph of g is a vertical stretch by a factor of 2, a refl ection in the
x-axis, and a translation 2 units left and 3 units up of the graph of f.
Transforming the Graph of y = a(x − h)2 + k
Consider function g in Example 3. Graph f (x) = g(x + 5).
SOLUTION
The function f is of the form y = g(x − h), where h = −5. So, the graph of f is a
horizontal translation 5 units left of the graph of g. To graph f, subtract 5 from the
x-coordinates of the points on the graph of g.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Graph the function. Compare the graph to the graph of f (x) = x2.
6. g(x) = 3(x − 1)2 + 6 7. h(x) = 1 —
2 (x + 4)2 − 2
8. Consider function g in Example 3. Graph f (x) = g(x) − 3.
x −4 −3
g(x) −5 1
Graphing f (x) = a(x − h)2 + k
Core Core ConceptConceptGraphing f (x) = a(x − h)2 + kThe vertex form of a quadratic function
is f (x) = a(x − h)2 + k, where a ≠ 0.
The graph of f (x) = a(x − h)2 + k is
a translation h units horizontally and
k units vertically of the graph of
f (x) = ax2.
The vertex of the graph of
f (x) = a(x − h)2 + k is (h, k),
and the axis of symmetry is x = h.
h
f(x) = ax2
y
x
f(x) = a(x − h)2 + k
k(h, k)
(0, 0)
−5
2
x2−4
y
f(x) = x2
y
g(x) = −2(x + 2)2 + 3
x
y
−4
2
−2−4−6
y
g(x) = −2(x + 2)2 + 3
f(x) = g(x + 5)
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Section 3.4 Graphing f (x) = a(x − h)2 + k 149
Modeling Real-Life Problems
Modeling with Mathematics
Water fountains are usually designed to give a specifi c visual effect. For example,
the water fountain shown consists of streams of water that are shaped like parabolas.
Notice how the streams are designed to land on the underwater spotlights. Write and
graph a quadratic function that models the path of a stream of water with a maximum
height of 5 feet, represented by a vertex of (3, 5), landing on a spotlight 6 feet from the
water jet, represented by (6, 0).
SOLUTION
1. Understand the Problem You know the vertex and another point on the graph
that represents the parabolic path. You are asked to write and graph a quadratic
function that models the path.
2. Make a Plan Use the given points and the vertex form to write a quadratic
function. Then graph the function.
3. Solve the ProblemUse the vertex form, vertex (3, 5), and point (6, 0) to fi nd the value of a.
f (x) = a(x − h)2 + k Write the vertex form of a quadratic function.
f (x) = a(x − 3)2 + 5 Substitute 3 for h and 5 for k.
0 = a(6 − 3)2 + 5 Substitute 6 for x and 0 for f (x).
0 = 9a + 5 Simplify.
− 5 —
9 = a Solve for a.
So, f (x) = − 5 —
9 (x − 3)2 + 5 models the path of a stream of water. Now graph
the function.
Step 1 Graph the axis of symmetry. Because h = 3, graph x = 3.
Step 2 Plot the vertex, (3, 5).
Step 3 Find and plot two more points on the graph. Because the x-axis represents
the water surface, the graph should only contain points with nonnegative
values of f (x). You know that (6, 0) is on the graph. To fi nd another point,
choose an x-value between x = 3 and x = 6. Then fi nd the corresponding
value of f (x).
f (4.5) = − 5 —
9 (4.5 − 3)2 + 5 = 3.75
So, plot (6, 0) and (4.5, 3.75).
Step 4 Refl ect the points plotted in Step 3
in the axis of symmetry. So, plot
(0, 0) and (1.5, 3.75).
Step 5 Draw a smooth curve through the points.
4. Look Back Use a graphing calculator to graph f (x) = − 5 —
9 (x − 3)2 + 5. Use the
maximum feature to verify that the maximum value is 5. Then use the zero feature
to verify that x = 6 is a zero of the function.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
9. WHAT IF? The vertex is (3, 6). Write and graph a quadratic function that models
the path.
W
t
N
g
h
w
0 2 4 6 x
y
0
2
4
6f(x) = − (x − 3)2 + 55
9
00
8
8MaximumX=3 Y=5
00
8
8ZeroX=6 Y=0
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150 Chapter 3 Graphing Quadratic Functions
Exercises3.4 Dynamic Solutions available at BigIdeasMath.com
1. VOCABULARY Compare the graph of an even function with the graph of an odd function.
2. OPEN-ENDED Write a quadratic function whose graph has a vertex of (1, 2).
3. WRITING Describe the transformation from the graph of f (x) = ax2 to the graph of g(x) = a(x − h)2 + k.
4. WHICH ONE DOESN’T BELONG? Which function does not belong with the other three?
Explain your reasoning.
f (x) = 2(x + 0)2f (x) = (x − 2)2 + 4 f (x) = 3(x + 1)2 + 1f (x) = 8(x + 4)2
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
In Exercises 5–12, determine whether the function is even, odd, or neither. (See Example 1.)
5. f (x) = 4x + 3 6. g(x) = 3x2
7. h(x) = 5x + 2 8. m(x) = 2x2 − 7x
9. p(x) = −x2 + 8 10. f (x) = − 1 — 2 x
11. n(x) = 2x2 − 7x + 3 12. r(x) = −6x2 + 5
In Exercises 13–18, determine whether the function represented by the graph is even, odd, or neither.
13.
1
3
x2−2
−2
y 14. 1 x
1−1−3
−3
−5
y
15.
2
4
x
−2
y 16. 1
x2−1
−2
−5
y
17.
2
x2−2
−2
y 18.
2
x2 4−2
y
In Exercises 19–22, fi nd the vertex and the axis of symmetry of the graph of the function.
19. f (x) = 3(x + 1)2 20. f (x) = 1 —
4 (x − 6)2
21. y = − 1 — 8 (x − 4)2 22. y = −5(x + 9)2
In Exercises 23–28, graph the function. Compare the graph to the graph of f (x) = x2. (See Example 2.)
23. g(x) = 2(x + 3)2 24. p(x) = 3(x − 1)2
25. r(x) = 1 —
4 (x + 10)2 26. n(x) =
1 —
3 (x − 6)2
27. d(x) = 1 —
5 (x − 5)2 28. q(x) = 6(x + 2)2
29. ERROR ANALYSIS Describe and correct the error in
determining whether the function f (x) = x2 + 3 is
even, odd, or neither.
f (x) = x2 + 3
f (−x) = (−x)2 + 3
= x2 + 3
= f (x)
So, f (x) is an odd function.
✗
30. ERROR ANALYSIS Describe and correct the error in
fi nding the vertex of the graph of the function.
y = −(x + 8)2
Because h = −8, the vertex
is (0, −8).
✗
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
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Section 3.4 Graphing f (x) = a(x − h)2 + k 151
In Exercises 31–34, fi nd the vertex and the axis of symmetry of the graph of the function.
31. y = −6(x + 4)2 − 3 32. f (x) = 3(x − 3)2 + 6
33. f (x) = −4(x + 3)2 + 1 34. y = −(x − 6)2 − 5
In Exercises 35–38, match the function with its graph.
35. y = −(x + 1)2 − 3 36. y = − 1 — 2 (x − 1)2 + 3
37. y = 1 —
3 (x − 1)2 + 3 38. y = 2(x + 1)2 − 3
A.
1
−1
−3
3
x1 3
y B.
2
−1x2−4
y
C.
−2
−4
−6
x2−4 −2
y D.
2
6
x2−2 4
y
In Exercises 39–44, graph the function. Compare the graph to the graph of f (x) = x2. (See Example 3.)
39. h(x) = (x − 2)2 + 4 40. g(x) = (x + 1)2 − 7
41. r(x) = 4(x − 1)2 − 5 42. n(x) = −(x + 4)2 + 2
43. g(x) = − 1 — 3 (x + 3)2 − 2 44. r(x) =
1 —
2 (x − 2)2 − 4
In Exercises 45–48, let f (x) = (x − 2)2 + 1. Match the function with its graph.
45. g(x) = f (x − 1) 46. r(x) = f (x + 2)
47. h(x) = f (x) + 2 48. p(x) = f (x) − 3
A.
2
4
x2 4 6
y B.
2
4
6
x2 4
y
C.
2
4
6
x2−2
y D.
−2
2
x2 4
y
In Exercises 49–54, graph g. (See Example 4.)
49. f (x) = 2(x − 1)2 + 1; g(x) = f (x + 3)
50. f (x) = −(x + 1)2 + 2; g(x) = 1 —
2 f (x)
51. f (x) = −3(x + 5)2 − 6; g(x) = 2f (x)
52. f (x) = 5(x − 3)2 − 1; g(x) = f (x) − 6
53. f (x) = (x + 3)2 + 5; g(x) = f (x − 4)
54. f (x) = −2(x − 4)2 − 8; g(x) = −f (x)
55. MODELING WITH MATHEMATICS The height
(in meters) of a bird diving to catch a fi sh is
represented by h(t) = 5(t − 2.5)2, where t is the
number of seconds after beginning the dive.
a. Graph h.
b. Another bird’s dive
is represented by
r(t) = 2h(t). Graph r.
c. Compare the graphs.
Which bird starts its
dive from a greater
height? Explain.
56. MODELING WITH MATHEMATICS A kicker punts
a football. The height (in yards) of the football is
represented by f (x) = − 1 —
9 (x − 30)2 + 25, where x
is the horizontal distance (in yards) from the kicker’s
goal line.
a. Graph f. Describe the domain and range.
b. On the next possession, the kicker punts the
football. The height of the football is represented
by g(x) = f (x + 5). Graph g. Describe the domain
and range.
c. Compare the graphs. On which possession does
the kicker punt closer to his goal line? Explain.
In Exercises 57–62, write a quadratic function in vertex form whose graph has the given vertex and passes through the given point.
57. vertex: (1, 2); passes through (3, 10)
58. vertex: (−3, 5); passes through (0, −14)
59. vertex: (−2, −4); passes through (−1, −6)
60. vertex: (1, 8); passes through (3, 12)
61. vertex: (5, −2); passes through (7, 0)
62. vertex: (−5, −1); passes through (−2, 2)
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152 Chapter 3 Graphing Quadratic Functions
63. MODELING WITH MATHEMATICS A portion of a
roller coaster track is in the shape of a parabola.
Write and graph a quadratic function that models this
portion of the roller coaster with a maximum height
of 90 feet, represented by a vertex of (25, 90), passing
through the point (50, 0). (See Example 5.)
64. MODELING WITH MATHEMATICS A fl are is launched
from a boat and travels in a parabolic path until
reaching the water. Write and graph a quadratic
function that models the path of the fl are with a
maximum height of 300 meters, represented by
a vertex of (59, 300), landing in the water at the
point (119, 0).
In Exercises 65–68, rewrite the quadratic function in vertex form.
65. y = 2x2 − 8x + 4 66. y = 3x2 + 6x − 1
67. f (x) = −5x2 + 10x + 3
68. f (x) = −x2 − 4x + 2
69. REASONING Can a function be symmetric about the
x-axis? Explain.
70. HOW DO YOU SEE IT? The graph of a quadratic
function is shown. Determine which symbols to
use to complete the vertex form of the quadratic
function. Explain your reasoning.
2
4
x2−6 −4 −2
−3
y
y = a(x 2)2 3
In Exercises 71–74, describe the transformation from the graph of f to the graph of h. Write an equation that represents h in terms of x.
71. f (x) = −(x + 1)2 − 2 72. f (x) = 2(x − 1)2 + 1
h(x) = f (x) + 4 h(x) = f (x − 5)
73. f (x) = 4(x − 2)2 + 3 74. f (x) = −(x + 5)2 − 6
h(x) = 2f (x) h(x) = 1 —
3 f (x)
75. REASONING The graph of y = x2 is translated 2 units
right and 5 units down. Write an equation for the
function in vertex form and in standard form. Describe
advantages of writing the function in each form.
76. THOUGHT PROVOKING Which of the following are
true? Justify your answers.
a. Any constant multiple of an even function is even.
b. Any constant multiple of an odd function is odd.
c. The sum or difference of two even functions is even.
d. The sum or difference of two odd functions is odd.
e. The sum or difference of an even function and an
odd function is odd.
77. COMPARING FUNCTIONS A cross section of a
birdbath can be modeled by y = 1 —
81 (x − 18)2 − 4,
where x and y are measured in inches. The graph
shows the cross section of another birdbath.
x
y
8 16 24 32
4
88 1616 2424
y
y = x2 − x 275
45
a. Which birdbath is deeper? Explain.
b. Which birdbath is wider? Explain.
78. REASONING Compare the graphs of y = 2x2 + 8x + 8
and y = x2 without graphing the functions. How can
factoring help you compare the parabolas? Explain.
79. MAKING AN ARGUMENT Your friend says all
absolute value functions are even because of their
symmetry. Is your friend correct? Explain.
Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencySolve the equation. (Section 2.4)
80. x(x − 1) = 0 81. (x + 3)(x − 8) = 0 82. (3x − 9)(4x + 12) = 0
Reviewing what you learned in previous grades and lessons
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Section 3.5 Graphing f (x) = a(x − p)(x − q) 153
Graphing f (x) = a(x − p)(x − q)3.5
Essential QuestionEssential Question What are some of the characteristics of the
graph of f (x) = a(x − p)(x − q)?
Using Zeros to Write Functions
Work with a partner. Each graph represents a function of the form
f (x) = (x − p)(x − q) or f (x) = −(x − p)(x − q). Write the function
represented by each graph. Explain your reasoning.
a.
−6
−4
4
6
b.
−6
−4
4
6
c.
−6
−4
4
6
d.
−6
−4
4
6
e.
−6
−4
4
6
f.
−6
−4
4
6
g.
−6
−4
4
6
h.
−6
−4
4
6
Communicate Your AnswerCommunicate Your Answer 2. What are some of the characteristics of the graph of f (x) = a(x − p)(x − q)?
3. Consider the graph of f (x) = a(x − p)(x − q).
a. Does changing the sign of a change the x-intercepts? Does changing the sign
of a change the y-intercept? Explain your reasoning.
b. Does changing the value of p change the x-intercepts? Does changing the
value of p change the y-intercept? Explain your reasoning.
CONSTRUCTINGVIABLEARGUMENTS
To be profi cient in math, you need to justify your conclusions and communicate them to others.
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154 Chapter 3 Graphing Quadratic Functions
3.5 Lesson What You Will LearnWhat You Will Learn Graph quadratic functions of the form f (x) = a(x − p)(x − q).
Use intercept form to fi nd zeros of functions.
Use characteristics to graph and write quadratic functions.
Graphing f (x) = a(x − p)(x − q)You have already graphed quadratic functions written in several different forms, such
as f (x) = ax2 + bx + c (standard form) and g(x) = a(x − h)2 + k (vertex form).
Quadratic functions can also be written in intercept form, f (x) = a(x − p)(x − q),
where a ≠ 0. In this form, the polynomial that defi nes a function is in factored form
and the x-intercepts of the graph can be easily determined.
Graphing f (x) = a(x − p)(x − q)
Graph f (x) = −(x + 1)(x − 5). Describe the domain and range.
SOLUTION
Step 1 Identify the x-intercepts. Because the x-intercepts are p = −1 and q = 5,
plot (−1, 0) and (5, 0).
Step 2 Find and graph the axis of symmetry.
x = p + q
— 2 =
−1 + 5 —
2 = 2
Step 3 Find and plot the vertex.
The x-coordinate of the vertex is 2.
To fi nd the y-coordinate of the vertex,
substitute 2 for x and simplify.
f (2) = −(2 + 1)(2 − 5) = 9
So, the vertex is (2, 9).
Step 4 Draw a parabola through the vertex and
the points where the x-intercepts occur.
The domain is all real numbers. The range is y ≤ 9.
intercept form, p. 154
Core VocabularyCore Vocabullarry
Core Core ConceptConceptGraphing f (x) = a(x − p)(x − q)• The x-intercepts are p and q.
• The axis of symmetry is halfway between
(p, 0) and (q, 0). So, the axis of symmetry
is x = p + q
— 2 .
• The graph opens up when a > 0, and the
graph opens down when a < 0.(q, 0)(p, 0)
x =
x
y
p + q2
x
y
4−2
10
6
8
(−1, 0) (5, 0)
(2, 9)
y1010
(2, 9),
f(x) = −(x + 1)(x − 5)
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Section 3.5 Graphing f (x) = a(x − p)(x − q) 155
Graphing a Quadratic Function
Graph f (x) = 2x2 − 8. Describe the domain and range.
SOLUTION
Step 1 Rewrite the quadratic function in intercept form.
f (x) = 2x2 − 8 Write the function.
= 2(x2 − 4) Factor out common factor.
= 2(x + 2)(x − 2) Difference of two squares pattern
Step 2 Identify the x-intercepts. Because the x-intercepts are p = −2 and q = 2,
plot (−2, 0) and (2, 0).
Step 3 Find and graph the axis of symmetry.
x = p + q
— 2 =
−2 + 2 —
2 = 0
Step 4 Find and plot the vertex.
The x-coordinate of the vertex is 0.
The y-coordinate of the vertex is
f (0) = 2(0)2 − 8 = −8.
So, the vertex is (0, −8).
Step 5 Draw a parabola through the vertex and
the points where the x-intercepts occur.
The domain is all real numbers. The range is y ≥ −8.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Describe the domain and range of the function.
1. f (x) = (x + 2)(x − 3) 2. g(x) = −2(x − 4)(x + 1) 3. h(x) = 4x2 − 36
Using Intercept Form to Find Zeros of FunctionsIn Section 3.2, you learned that a zero of a function is an x-value for which f (x) = 0.
You can use the intercept form of a function to fi nd the zeros of the function.
Finding Zeros of a Function
Find the zeros of f (x) = (x − 1)(x + 2).
SOLUTION
To fi nd the zeros, determine the x-values for which f (x) is 0.
f (x) = (x − 1)(x + 2) Write the function.
0 = (x − 1)(x + 2) Substitute 0 for f(x).
x − 1 = 0 or x + 2 = 0 Zero-Product Property
x = 1 or x = −2 Solve for x.
So, the zeros of the function are −2 and 1.
REMEMBERFunctions have zeros, and graphs have x-intercepts.
Check
x
y
4−4
(−2, 0) (2, 0)
(0, −8)
−2
−4
f(x) = 2x2 − 8
−5
−4
6
4
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156 Chapter 3 Graphing Quadratic Functions
Finding Zeros of Functions
Find the zeros of each function.
a. h(x) = x2 − 16 b. f (x) = −2x2 − 10x − 12
SOLUTION
Write each function in intercept form to identify the zeros.
a. h(x) = x2 − 16 Write the function.
= (x + 4)(x − 4) Difference of two squares pattern
So, the zeros of the function are −4 and 4.
b. f (x) = −2x2 − 10x − 12 Write the function.
= −2(x2 + 5x + 6) Factor out common factor.
= −2(x + 3)(x + 2) Factor the trinomial.
So, the zeros of the function are −3 and −2.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Find the zero(s) of the function.
4. f (x) = (x − 6)(x − 1) 5. h(x) = x2 − 1 6. g(x) = 3x2 − 12x + 12
Using Characteristics to Graph and Write Quadratic Functions
Graphing a Quadratic Function Using Zeros
Use zeros to graph h(x) = x2 − 2x − 3.
SOLUTION
The function is in standard form. You know that the parabola opens up (a > 0) and
the y-intercept is −3. So, begin by plotting (0, −3).
Notice that the polynomial that defi nes the
function is factorable. So, write the function
in intercept form and identify the zeros.
h(x) = x2 − 2x − 3 Write the function.
= (x + 1)(x − 3) Factor the trinomial.
The zeros of the function are −1 and 3. So,
plot (−1, 0) and (3, 0). Draw a parabola through
the points.
ATTENDING TO PRECISION
To sketch a more precise graph, make a table of values and plot other points on the graph.
Core Core ConceptConceptFactors and ZerosFor any factor x − n of a polynomial, n is a zero of the function defi ned by
the polynomial.
x
y2
−4
42−2
h(x) = x2 − 2x − 3
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Section 3.5 Graphing f (x) = a(x − p)(x − q) 157
Writing Quadratic Functions
Write a quadratic function in standard form whose graph satisfi es the
given condition(s).
a. vertex: (−3, 4)
b. passes through (−9, 0), (−2, 0), and (−4, 20)
SOLUTION
a. Because you know the vertex, use vertex form to write a function.
f (x) = a(x − h)2 + k Vertex form
= 1(x + 3)2 + 4 Substitute for a, h, and k.
= x2 + 6x + 9 + 4 Find the product (x + 3)2.
= x2 + 6x + 13 Combine like terms.
b. The given points indicate that the x-intercepts are −9 and −2. So, use intercept
form to write a function.
f (x) = a(x − p)(x − q) Intercept form
= a(x + 9)(x + 2) Substitute for p and q.
Use the other given point, (−4, 20), to fi nd the value of a.
20 = a(−4 + 9)(−4 + 2) Substitute −4 for x and 20 for f(x).
20 = a(5)(−2) Simplify.
−2 = a Solve for a.
Use the value of a to write the function.
f (x) = −2(x + 9)(x + 2) Substitute −2 for a.
= −2x2 − 22x − 36 Simplify.
Check
2
−70
−14
40
Y=20
14
4f(x) = −2x2 − 22x − 36
X=-4
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Use zeros to graph the function.
7. f (x) = (x − 1)(x − 4) 8. g(x) = x2 + x − 12
Write a quadratic function in standard form whose graph satisfi es the given condition(s).
9. x-intercepts: −1 and 1 10. vertex: (8, 8)
11. passes through (0, 0), (10, 0), and (4, 12)
12. passes through (−5, 0), (4, 0), and (3, −16)
STUDY TIPIn part (a), many possible functions satisfy the given condition. The value a can be any nonzero number. To allow easier calculations, let a = 1. By letting a = 2, the resulting function would be f (x) = 2x2 + 12x + 22.
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158 Chapter 3 Graphing Quadratic Functions
Exercises3.5 Dynamic Solutions available at BigIdeasMath.com
In Exercises 3–6, fi nd the x-intercepts and axis of symmetry of the graph of the function.
3. y = (x − 1)(x + 3)
2 x
y
−2−4
1
−2
4.
5. f (x) = −5(x + 7)(x − 5) 6. g(x) = 2 —
3 x(x + 8)
In Exercises 7–12, graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Describe the domain and range of the function. (See Example 1.)
7. f (x) = (x + 4)(x + 1) 8. y = (x − 2)(x + 2)
9. y = −(x + 6)(x − 4) 10. h(x) = −4(x − 7)(x − 3)
11. g(x) = 5(x + 1)(x + 2) 12. y = −2(x − 3)(x + 4)
In Exercises 13–20, graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Describe the domain and range of the function. (See Example 2.)
13. y = x2 − 9 14. f (x) = x2 − 8x
15. h(x) = −5x2 + 5x 16. y = 3x2 − 48
17. q(x) = x2 + 9x + 14 18. p(x) = x2 + 6x − 27
19. y = 4x2 − 36x + 32 20. y = −2x2 − 4x + 30
In Exercises 21–28, fi nd the zero(s) of the function. (See Examples 3 and 4.)
21. y = −2(x − 2)(x − 10) 22. f (x) = 1 —
3 (x + 5)(x − 1)
23. f (x) = x2 − 4 24. h(x) = x2 − 36
25. g(x) = x2 + 5x − 24 26. y = x2 − 17x + 52
27. y = 3x2 − 15x − 42 28. g(x) = −4x2 − 8x − 4
In Exercises 29–34, match the function with its graph.
29. y = (x + 5)(x + 3) 30. y = (x + 5)(x − 3)
31. y = (x − 5)(x + 3) 32. y = (x − 5)(x − 3)
33. y = (x + 5)(x − 5) 34. y = (x + 3)(x − 3)
A.
x
y
−2−6 2 6
4
−8
−28
B.
x
y
62
4
8
12
C.
x
y
−18
−4 4 8
6 D.
x
y
−2−6
4
8
E.
x
y4
−4
−12
2 4
F. y6
−6
−18
x4−8
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. COMPLETE THE SENTENCE The values p and q are __________ of the graph of the function
f (x) = a(x − p)(x − q).
2. WRITING Explain how to fi nd the maximum value or minimum value of a quadratic function
when the function is given in intercept form.
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
x
y
2
4
4 6
y
4
y = −2(x − 2)(x − 5)
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Section 3.5 Graphing f (x) = a(x − p)(x − q) 159
In Exercises 35–40, use zeros to graph the function. (See Example 5.)
35. f (x) = (x + 2)(x − 6) 36. g(x) = −3(x + 1)(x + 7)
37. y = x2 − 11x + 18 38. y = x2 − x − 30
39. y = −5x2 − 10x + 40 40. h(x) = 8x2 − 8
ERROR ANALYSIS In Exercises 41 and 42, describe and correct the error in fi nding the zeros of the function.
41. y = 5(x + 3)(x − 2)
The zeros of the function are
3 and −2.
✗
42.
y = x2 − 9
The zero of the function is 9.✗
In Exercises 43–54, write a quadratic function in standard form whose graph satisfi es the given condition(s). (See Example 6.)
43. vertex: (7, −3) 44. vertex: (4, 8)
45. x-intercepts: 1 and 9 46. x-intercepts: −2 and −5
47. passes through (−4, 0), (3, 0), and (2, −18)
48. passes through (−5, 0), (−1, 0), and (−4, 3)
49. passes through (7, 0)
50. passes through (0, 0) and (6, 0)
51. axis of symmetry: x = −5
52. y increases as x increases when x < 4; y decreases as
x increases when x > 4.
53. range: y ≥ −3 54. range: y ≤ 10
In Exercises 55–58, write the quadratic function represented by the graph.
55.
−4 1 3 x
y2
−4
−8(−3, 0)
(1, −8)
(2, 0)
56.
(1, 0) (7, 0)
(6, 5)4
8
y
x2 4 6
57. (2, 32)
(4, 0)
(−2, 0)
14
35
−14
y
x1−3 3
58.
(6, 0)
(10, 0)
(8, −2)
2
−2
4
y
x4 8
In Exercises 59 and 60, all the zeros of a function are given. Use the zeros and the other point given to write a quadratic function represented by the table.
59. x y
0 0
2 30
7 0
60. x y
−3 0
1 −72
4 0
In Exercises 61–64, sketch a parabola that satisfi es the given conditions.
61. x-intercepts: −4 and 2; range: y ≥ −3
62. axis of symmetry: x = 6; passes through (4, 15)
63. range: y ≤ 5; passes through (0, 2)
64. x-intercept: 6; y-intercept: 1; range: y ≥ −4
65. MODELING WITH MATHEMATICS Satellite dishes are
shaped like parabolas to optimally receive signals.
The cross section of a satellite dish can be modeled
by the function shown, where x and y are measured in
feet. The x-axis represents the top of the opening of
the dish.
x
y
2−2
−1
x222
y = (x2 − 4)18
a. How wide is the
satellite dish?
b. How deep is the
satellite dish?
c. Write a quadratic
function in standard
form that models the
cross section of a
satellite dish that is
6 feet wide and
1.5 feet deep.
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160 Chapter 3 Graphing Quadratic Functions
66. MODELING WITH MATHEMATICS A professional
basketball player’s shot is modeled by the function
shown, where x and y are measured in feet.
5 10 15 20 x
y
−5
−10
y = − (x2 − 19x + 48)120
(3, 0)
a. Does the player make the shot? Explain.
b. The basketball player releases another shot from
the point (13, 0) and makes the shot. The shot
also passes through the point (10, 1.4). Write a
quadratic function in standard form that models
the path of the shot.
USING STRUCTURE In Exercises 67–70, match the function with its graph.
67. y = −x2 + 5x 68. y = x2 − x − 12
69. y = −2x2 − 8 70. y = 3x2 + 6
A.
x
y B.x
y
C.
x
y D.
x
y
71. CRITICAL THINKING Write a quadratic function
represented by the table, if possible. If not,
explain why.
x −5 −3 −1 1
y 0 12 4 0
72. HOW DO YOU SEE IT? The graph shows the
parabolic arch that supports the roof of a convention
center, where x and y are measured in feet.
50 100 150 200 250 300 350 400 x
y
−50
50
a. The arch can be represented by a function of the
form f (x) = a(x − p)(x − q). Estimate the values
of p and q.
b. Estimate the width and height of the arch.
Explain how you can use your height estimate
to calculate a.
ANALYZING EQUATIONS In Exercises 73 and 74, (a) rewrite the quadratic function in intercept form and (b) graph the function using any method. Explain the method you used.
73. f (x) = −3(x + 1)2 + 27
74. g(x) = 2(x − 1)2 − 2
75. WRITING Can a quadratic function with exactly one
real zero be written in intercept form? Explain.
76. MAKING AN ARGUMENT Your friend claims that any
quadratic function can be written in standard form and
in vertex form. Is your friend correct? Explain.
77. REASONING Let k be a constant. Find the zeros of the
function f (x) = kx2 − k2x − 2k3 in terms of k.
78. THOUGHT PROVOKING Sketch the graph of each
function. Explain your procedure.
a. f (x) = (x2 − 1)(x2 − 4)
b. g(x) = x(x2 − 1)(x2 − 4)
PROBLEM SOLVING In Exercises 79 and 80, write two quadratic functions whose graphs intersect at the given points. Explain your reasoning.
79. (−4, 0) and (2, 0) 80. (3, 6) and (7, 6)
Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyFind the distance between the two points. (Skills Review Handbook)
81. (2, 1) and (3, 0) 82. (−1, 8) and (−9, 2) 83. (−1, −6) and (−3, −5)
Reviewing what you learned in previous grades and lessons
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Section 3.6 Focus of a Parabola 161
Focus of a Parabola3.6
Essential QuestionEssential Question What is the focus of a parabola?
Analyzing Satellite Dishes
Work with a partner. Vertical rays enter a satellite dish whose cross section is a
parabola. When the rays hit the parabola, they refl ect at the same angle at which they
entered. (See Ray 1 in the fi gure.)
a. Draw the refl ected rays so that they intersect the y-axis.
b. What do the refl ected rays have in common?
c. The optimal location for the receiver of the satellite dish is at a point called the
focus of the parabola. Determine the location of the focus. Explain why this makes
sense in this situation.
−1−2 1 2
1
2
y = x214
x
yRay Ray Ray
incoming angle outgoingangle
Analyzing Spotlights
Work with a partner. Beams of light are coming from the bulb in a spotlight, located
at the focus of the parabola. When the beams hit the parabola, they refl ect at the same
angle at which they hit. (See Beam 1 in the fi gure.) Draw the refl ected beams. What do
they have in common? Would you consider this to be the optimal result? Explain.
−1−2 1 2
1
2 y = x212
x
y
Beam
Beam
Beam
bulb
incoming angle
outgoingangle
Communicate Your AnswerCommunicate Your Answer 3. What is the focus of a parabola?
4. Describe some of the properties of the focus of a parabola.
CONSTRUCTING VIABLE ARGUMENTS
To be profi cient in math, you need to make conjectures and build logical progressions of statements to explore the truth of your conjectures.
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162 Chapter 3 Graphing Quadratic Functions
3.6 Lesson What You Will LearnWhat You Will Learn Explore the focus and the directrix of a parabola.
Write equations of parabolas.
Solve real-life problems.
Exploring the Focus and DirectrixPreviously, you learned that the graph of a quadratic function is a parabola that opens
up or down. A parabola can also be defi ned as the set of all points (x, y) in a plane that
are equidistant from a fi xed point called the focus and a fi xed line called the directrix.
axis ofsymmetry
The directrix isperpendicular to theaxis of symmetry.
The focus is in the interiorof the parabola and lies onthe axis of symmetry.
The vertex lies halfwaybetween the focus andthe directrix.
focus, p. 162directrix, p. 162
PreviousperpendicularDistance Formulacongruent
Core VocabularyCore Vocabullarry
Using the Distance Formula to Write an Equation
Use the Distance Formula to write an equation of the
parabola with focus F(0, 2) and directrix y = −2.
SOLUTION
Notice the line segments drawn from point F to point P
and from point P to point D. By the defi nition of a
parabola, these line segments must be congruent.
PD = PF Defi nition of a parabola
√——
(x − x1)2 + (y − y1)
2 = √——
(x − x2)2 + (y − y2)
2 Distance Formula
√——
(x − x)2 + (y − (−2))2 = √——
(x − 0)2 + (y − 2)2 Substitute for x1, y1, x2, and y2.
√—
(y + 2)2 = √——
x2 + (y − 2)2 Simplify.
(y + 2)2 = x2 + (y − 2)2 Square each side.
y2 + 4y + 4 = x2 + y2 − 4y + 4 Expand.
8y = x2 Combine like terms.
y = 1 —
8 x2 Divide each side by 8.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
1. Use the Distance Formula to write an equation
of the parabola with focus F(0, −3) and
directrix y = 3.
REMEMBERSquaring a positive number and fi nding a square root are inverse operations. So, you can square each side of the equation to eliminate the radicals.
REMEMBERThe distance from a point to a line is defi ned as the length of the perpendicular segment from the point to the line.
x
y
F(0, 2)
P(x, y)
D(x, −2)y = −2
x
y
F(0, −3)P(x, y)
D(x, 3)
y = 3
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Section 3.6 Focus of a Parabola 163
You can derive the equation of a parabola that opens up or down with vertex (0, 0),
focus (0, p), and directrix y = −p using the procedure in Example 1.
√——
(x − x)2 + (y − (−p))2 = √——
(x − 0)2 + (y − p)2
(y + p)2 = x2 + (y − p)2
y2 + 2py + p2 = x2 + y2 − 2py + p2
4py = x2
y = 1 —
4p x2
The focus and directrix each lie ∣ p ∣ units from the vertex. Parabolas can also open left
or right, in which case the equation has the form x = 1 —
4p y2 when the vertex is (0, 0).
LOOKING FOR STRUCTURE
Notice that y = 1 — 4p
x2 is
of the form y = ax2. So, changing the value of p vertically stretches or shrinks the parabola.
STUDY TIPNotice that parabolas opening left or right do not represent functions.
Core Core ConceptConceptStandard Equations of a Parabola with Vertex at the OriginVertical axis of symmetry (x = 0)
Equation: y = 1 —
4p x2
x
yfocus:(0, p)
directrix:y = −p
vertex: (0, 0)
x
y
focus:(0, p)
directrix:y = −p
vertex: (0, 0)Focus: (0, p)
Directrix: y = −p
p > 0 p < 0
Horizontal axis of symmetry (y = 0)
Equation: x = 1 —
4p y2
x
y
focus:(p, 0)
directrix:x = −p
vertex:(0, 0)
x
y
focus:(p, 0)
directrix:x = −p
vertex:(0, 0)
Focus: (p, 0)
Directrix: x = −p
p > 0 p < 0
Graphing an Equation of a Parabola
Identify the focus, directrix, and axis of symmetry of −4x = y2. Graph the equation.
SOLUTION
Step 1 Rewrite the equation in standard form.
−4x = y2 Write the original equation.
x = − 1 —
4 y2 Divide each side by –4.
Step 2 Identify the focus, directrix, and axis of symmetry. The equation has the form
x = 1 —
4p y2, where p = −1. The focus is (p, 0), or (−1, 0). The directrix is
x = −p, or x = 1. Because y is squared, the axis of symmetry is the x-axis.
Step 3 Use a table of values to graph the
equation. Notice that it is easier to
substitute y-values and solve for x.
Opposite y-values result in the
same x-value.
x
y
F(0, p)
P(x, y)
D(x, −p)y = −p
x
y4
−4
2−2−4(−1, 0)
x = 1
y 0 ±1 ±2 ±3 ±4
x 0 −0.25 −1 −2.25 −4
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164 Chapter 3 Graphing Quadratic Functions
Writing an Equation of a Parabola
Write an equation of the parabola shown.
SOLUTION
Because the vertex is at the origin and the axis of symmetry is vertical, the equation
has the form y = 1 —
4p x2. The directrix is y = −p = 3, so p = −3. Substitute −3 for p to
write an equation of the parabola.
y = 1 —
4(−3) x2 = −
1 —
12 x2
So, an equation of the parabola is y = − 1 —
12 x2.
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Identify the focus, directrix, and axis of symmetry of the parabola. Then graph the equation.
2. y = 0.5x2 3. −y = x2 4. y2 = 6x
Write an equation of the parabola with vertex at (0, 0) and the given directrix or focus.
5. directrix: x = −3 6. focus: (−2, 0) 7. focus: ( 0, 3 —
2 )
The vertex of a parabola is not always at the origin. As in previous transformations,
adding a value to the input or output of a function translates its graph.
Writing Equations of Parabolas
Core Core ConceptConceptStandard Equations of a Parabola with Vertex at (h, k)Vertical axis of symmetry (x = h)
Equation: y = 1 —
4p (x − h)2 + k
x
y
(h, k)
(h, k + p)x = h
y = k − p
xyx = h
y = k − p
(h, k)
(h, k + p)
Focus: (h, k + p)
Directrix: y = k − p
p > 0 p < 0
Horizontal axis of symmetry (y = k)
Equation: x = 1 —
4p (y − k)2 + h
x
y
y = k
x = h − p
(h, k)
(h + p, k)
x
y
y = k
x = h − p
(h, k)
(h + p, k)Focus: (h + p, k)
Directrix: x = h − p
p > 0 p < 0
STUDY TIPThe standard form for a vertical axis of symmetry looks like vertex form. To remember the standard form for a horizontal axis of symmetry, switch x and y, and h and k.
x
y4
−2
4−4
directrix
vertex
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Section 3.6 Focus of a Parabola 165
Writing an Equation of a Translated Parabola
Write an equation of the parabola shown.
SOLUTION
Because the vertex is not at the origin and the axis of symmetry is horizontal, the
equation has the form x = 1 —
4p (y − k)2 + h. The vertex (h, k) is (6, 2) and the focus
(h + p, k) is (10, 2), so h = 6, k = 2, and p = 4. Substitute these values to write an
equation of the parabola.
x = 1 —
4(4) (y − 2)2 + 6 =
1 —
16 (y − 2)2 + 6
So, an equation of the parabola is x = 1 —
16 (y − 2)2 + 6.
Solving a Real-Life Problem
An electricity-generating dish uses a parabolic refl ector to concentrate sunlight onto a
high-frequency engine located at the focus of the refl ector. The sunlight heats helium
to 650°C to power the engine. Write an equation that represents the cross section of the
dish shown with its vertex at (0, 0). What is the depth of the dish?
SOLUTION
Because the vertex is at the origin, and the axis of symmetry is vertical, the equation
has the form y = 1 —
4p x2. The engine is at the focus, which is 4.5 meters above the
vertex. So, p = 4.5. Substitute 4.5 for p to write the equation.
y = 1 —
4(4.5) x2 =
1 —
18 x2
The depth of the dish is the y-value at the dish’s outside edge. The dish extends
8.5 —
2 = 4.25 meters to either side of the vertex (0, 0), so fi nd y when x = 4.25.
y = 1 —
18 (4.25)2 ≈ 1
The depth of the dish is about 1 meter.
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8. Write an equation of a parabola with vertex (−1, 4) and focus (−1, 2).
9. A parabolic microwave antenna is 16 feet in diameter. Write an equation that
represents the cross section of the antenna with its vertex at (0, 0) and its focus
10 feet to the right of the vertex. What is the depth of the antenna?
x
y
4
8
4 12 16
vertex focus
Solving Real-Life ProblemsParabolic refl ectors have cross
sections that are parabolas.
Incoming sound, light, or other
energy that arrives at a parabolic
refl ector parallel to the axis of
symmetry is directed to the focus
(Diagram 1). Similarly, energy that is emitted from the focus of a parabolic refl ector
and then strikes the refl ector is directed parallel to the axis of symmetry (Diagram 2).
Diagram 1
Focus
Diagram 2
Focus
x
y
4.5 m
8.5 m
engine
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166 Chapter 3 Graphing Quadratic Functions
Exercises3.6 Dynamic Solutions available at BigIdeasMath.com
In Exercises 3–10, use the Distance Formula to write an equation of the parabola. (See Example 1.)
3. 4.
5. focus: (0, −2) 6. directrix: y = 7
directrix: y = 2 focus: (0, −7)
7. vertex: (0, 0) 8. vertex: (0, 0)
directrix: y = −6 focus: (0, 5)
9. vertex: (0, 0) 10. vertex: (0, 0)
focus: (0, −10) directrix: y = −9
11. ANALYZING RELATIONSHIPS Which of the given
characteristics describe parabolas that open down?
Explain your reasoning.
○A focus: (0, −6) ○B focus: (0, −2)
directrix: y = 6 directrix: y = 2
○C focus: (0, 6) ○D focus: (0, −1)
directrix: y = −6 directrix: y = 1
12. REASONING Which
of the following are
possible coordinates of
the point P in the graph
shown? Explain.
○A (−6, −1) ○B ( 3, − 1 —
4 ) ○C ( 4, −
4 —
9 )
○D ( 1, 1 —
36 ) ○E (6, −1) ○F ( 2, −
1 —
18 )
In Exercises 13–20, identify the focus, directrix, and axis of symmetry of the parabola. Graph the equation. (See Example 2.)
13. y = 1 —
8 x2 14. y = −
1 —
12 x2
15. x = − 1 —
20 y2 16. x =
1 —
24 y2
17. y2 = 16x 18. −x2 = 48y
19. 6x2 + 3y = 0 20. 8x2 − y = 0
ERROR ANALYSIS In Exercises 21 and 22, describe and correct the error in graphing the parabola.
21. –6x + y2 = 0
x
y
4
8
4
(0, 1.5)
−4 y = −1.5
✗
22. 0.5y2 + x = 0
x
y
2
2 4(0.5, 0)
−2−4
x = −0.5
✗
23. ANALYZING EQUATIONS The cross section (with
units in inches) of a parabolic satellite dish can be
modeled by the equation y = 1 —
38 x2. How far is the
receiver from the vertex of the cross section? Explain.
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. COMPLETE THE SENTENCE A parabola is the set of all points in a plane equidistant from a fi xed point
called the ______ and a fi xed line called the __________ .
2. WRITING Explain how to fi nd the coordinates of the focus of a parabola with vertex ( 0, 0 ) and
directrix y = 5.
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
x
D(x, −1)
P(x, y)F(0, 1)
y
y = −1
D(x, 4)
P(x, y)
F(0, −4)
x
y
y = 4
x
y
P(x, y)
V(0, 0)
F(0, −9)
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Section 3.6 Focus of a Parabola 167
24. ANALYZING EQUATIONS The cross section (with
units in inches) of a parabolic spotlight can be
modeled by the equation x = 1 —
20 y2. How far is the bulb
from the vertex of the cross section? Explain.
In Exercises 25–28, write an equation of the parabola shown. (See Example 3.)
25.
x
y = −8
y
directrix
vertex
26.
27. 28.
In Exercises 29–36, write an equation of the parabola with the given characteristics.
29. focus: (3, 0) 30. focus: ( 2 — 3 , 0 )
directrix: x = −3 directrix: x = − 2 —
3
31. directrix: x = −10 32. directrix: y = 8 —
3
vertex: (0, 0) vertex: (0, 0)
33. focus: ( 0, − 5 —
3 ) 34. focus: ( 0,
5 —
4 )
directrix: y = 5 —
3 directrix: y = −
5 —
4
35. focus: ( 0, 6 —
7 ) 36. focus: ( −
4 —
5 , 0 )
vertex: (0, 0) vertex: (0, 0)
In Exercises 37–40, write an equation of the parabola shown. (See Example 4.)
37. 38.
x
y
4
8
vertexfocus
−4
−8
−12
x
y
2
4
2 6
vertex focus
−2
39. 40.
x
y
2
3
−1−2 1 2
vertex
focus
x
y
focus
−2 2−6vertex
−10
−14
−10
In Exercises 41–46, identify the vertex, focus, directrix, and axis of symmetry of the parabola. Describe the transformations of the graph of the standard equation with p = 1 and vertex (0, 0).
41. y = 1 —
8 (x − 3)2 + 2 42. y = −
1 —
4 (x + 2)2 + 1
43. x = 1 —
16 (y − 3)2 + 1 44. y = (x + 3)2 − 5
45. x = −3(y + 4)2 + 2 46. x = 4(y + 5)2 − 1
47. MODELING WITH MATHEMATICS Scientists studying
dolphin echolocation simulate the projection of a
bottlenose dolphin’s clicking sounds using computer
models. The models originate the sounds at the focus
of a parabolic refl ector. The parabola in the graph
shows the cross section of the refl ector with focal
length of 1.3 inches and aperture width of 8 inches.
Write an equation to represent the cross section
of the refl ector. What is the depth of the refl ector?
(See Example 5.)
x
y
focal length
aperture
F
x
y = y
directrix
vertex
34
x
x =
y
directrix
vertex
52
x
x = −2
y
directrix
vertex
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168 Chapter 3 Graphing Quadratic Functions
Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyShow that an exponential model fi ts the data. Then write a recursive rule that models the data. (Section 1.6)
55. n 0 1 2 3 4 5
f (n) 0.25 0.5 1 2 4 8
57. n 0 1 2 3 4 5
f (n) 144 72 36 18 9 4.5
56. n 0 1 2 3 4 5
f (n) 2 6 18 54 162 486
58. n 0 1 2 3 4 5
f (n) 1250 250 50 10 2 2 —
5
Reviewing what you learned in previous grades and lessons
48. MODELING WITH MATHEMATICS Solar energy can be
concentrated using long troughs that have a parabolic
cross section as shown in the fi gure. Write an equation
to represent the cross section of the trough. What are
the domain and range in this situation? What do
they represent?
1.7 m
5.8 m
49. ABSTRACT REASONING As ∣ p ∣ increases, how does
the width of the graph of the equation y = 1 —
4p x2
change? Explain your reasoning.
50. HOW DO YOU SEE IT? The graph shows the path of a
volleyball served from an initial height of 6 feet as it
travels over a net.
x
y A
B
C
a. Label the vertex, focus, and a point on
the directrix.
b. An underhand serve follows the same parabolic
path but is hit from a height of 3 feet. How does
this affect the focus? the directrix?
51. CRITICAL THINKING The distance from point P to the
directrix is 2 units. Write an equation of the parabola.
x
y
P(−2, 1)
V(0, 0)
52. THOUGHT PROVOKING Two parabolas have the
same focus (a, b) and focal length of 2 units. Write
an equation of each parabola. Identify the directrix of
each parabola.
53. REPEATED REASONING Use the Distance Formula
to derive the equation of a parabola that opens to
the right with vertex (0, 0), focus (p, 0), and
directrix x = −p.
x
y
F(p, 0)
P(x, y)D(−p, y)
x = −p
54. PROBLEM SOLVING The latus rectum of a parabola is
the line segment that is parallel to the directrix, passes
through the focus, and has endpoints that lie on the
parabola. Find the length of the latus rectum of the
parabola shown.
x
y
V(0, 0)
F(0, 2)A B
latusrectum
y = −2
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Section 3.7 Comparing Linear, Exponential, and Quadratic Functions 169
Essential QuestionEssential Question How can you compare the growth rates of
linear, exponential, and quadratic functions?
Comparing Speeds
Work with a partner. Three cars start traveling at the same time. The distance
traveled in t minutes is y miles. Complete each table and sketch all three graphs in the
same coordinate plane. Compare the speeds of the three cars. Which car has a constant
speed? Which car is accelerating the most? Explain your reasoning.
t y = t
0
0.2
0.4
0.6
0.8
1.0
t y = 2t − 1
0
0.2
0.4
0.6
0.8
1.0
t y = t2
0
0.2
0.4
0.6
0.8
1.0
Comparing Speeds
Work with a partner. Analyze the speeds of the three cars over the given time
periods. The distance traveled in t minutes is y miles. Which car eventually overtakes
the others?
t y = t
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
t y = 2t − 1
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
t y = t2
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Communicate Your AnswerCommunicate Your Answer 3. How can you compare the growth rates of linear, exponential, and quadratic
functions?
4. Which function has a growth rate that is eventually much greater than the growth
rates of the other two functions? Explain your reasoning.
COMPARING PREDICTIONS
To be profi cient in math, you need to visualize the results of varying assumptions, explore consequences, and compare predictions with data.
Comparing Linear, Exponential, and Quadratic Functions
3.7
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170 Chapter 3 Graphing Quadratic Functions
3.7 Lesson What You Will LearnWhat You Will Learn Choose functions to model data.
Write functions to model data.
Compare functions using average rates of change.
Choosing Functions to Model DataSo far, you have studied linear functions, exponential functions, and quadratic
functions. You can use these functions to model data.
Using Graphs to Identify Functions
Plot the points. Tell whether the points appear to represent a linear, an exponential, or
a quadratic function.
a. (4, 4), (2, 0), (0, 0), b. (0, 1), (2, 4), (4, 7), c. (0, 2), (2, 8), (1, 4),
( 1, − 1 —
2 ) , (−2, 4) (−2, −2), (−4, −5) (−1, 1), ( −2,
1 —
2 )
SOLUTION
a.
−2
2
4
x42−2
y b.
−4
−8
4
x4−4
8y c.
4
6
8
x2−2
y
quadratic linear exponential
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Plot the points. Tell whether the points appear to represent a linear, an exponential, or a quadratic function.
1. (−1, 5), (2, −1), (0, −1), (3, 5), (1, −3)
2. (−1, 2), (−2, 8), (−3, 32), ( 0, 1 —
2 ) , ( 1,
1 —
8 )
3. (−3, 5), (0, −1), (2, −5), (−4, 7), (1, −3)
Previousaverage rate of change slope
Core VocabularyCore Vocabullarry
Core Core ConceptConceptLinear, Exponential, and Quadratic Functions Linear Function Exponential Function Quadratic Function
y = mx + b y = abx y = ax2 + bx + c
x
y
x
y
x
y
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Section 3.7 Comparing Linear, Exponential, and Quadratic Functions 171
Using Differences or Ratios to Identify Functions
Tell whether each table of values represents a linear, an exponential, or a
quadratic function.
a. x −3 −2 −1 0 1
y 11 8 5 2 −1
b. x −2 −1 0 1 2
y 1 2 4 8 16
c. x −2 −1 0 1 2
y −1 −2 −1 2 7
SOLUTION
a. + 1
− 3
+ 1
− 3
+ 1 + 1
− 3 − 3
x −3 −2 −1 0 1
y 11 8 5 2 −1
b. + 1
× 2
+ 1
× 2
+ 1 + 1
× 2 × 2
x −2 −1 0 1 2
y 1 2 4 8 16
The fi rst differences are constant.
So, the table represents a linear
function.
Consecutive y-values have a
common ratio. So, the table
represents an exponential function.
c. + 1
− 1
+ 1
+ 1
+ 2
+ 1 + 1
+ 3
+ 2
+ 5
+ 2
x −2 −1 0 1 2
y −1 −2 −1 2 7
first differences
second differences
The second differences are
constant. So, the table represents
a quadratic function.
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4. Tell whether the table of values represents a linear, an exponential, or a
quadratic function.
STUDY TIPFirst determine that the differences of consecutive x-values are constant. Then check whether the fi rst differences are constant or consecutive y-values have a common ratio. If neither of these is true, check whether the second differences are constant.
x −1 0 1 2 3
y 1 3 9 27 81
Core Core ConceptConceptDifferences and Ratios of FunctionsYou can use patterns between consecutive data pairs to determine which type of
function models the data. The differences of consecutive y-values are called fi rst differences. The differences of consecutive fi rst differences are called second differences.
• Linear Function The fi rst differences are constant.
• Exponential Function Consecutive y-values have a common ratio.
• Quadratic Function The second differences are constant.
In all cases, the differences of consecutive x-values need to be constant.
STUDY TIPThe fi rst differences for exponential and quadratic functions are not constant.
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172 Chapter 3 Graphing Quadratic Functions
Writing Functions to Model Data
Writing a Function to Model Data
Tell whether the table of values represents a linear, an exponential, or a quadratic
function. Then write the function.
SOLUTION
Step 1 Determine which type
of function the table of
values represents.
The second differences
are constant. So, the
table represents a
quadratic function.
Step 2 Write an equation of the quadratic function. Using the table, notice that the
x-intercepts are 1 and 3. So, use intercept form to write a function.
f(x) = a(x − 1)(x − 3) Substitute for p and q in intercept form.
Use another point from the table, such as (0, 12), to fi nd a.
12 = a(0 − 1)(0 − 3) Substitute 0 for x and 12 for f(x).
4 = a Solve for a.
Use the value of a to write the function.
f(x) = 4(x − 1)(x − 3) Substitute 4 for a.
= 4x2 − 16x + 12 Use the FOIL Method and combine like terms.
So, the quadratic function is f(x) = 4x2 − 16x + 12.
Previously, you wrote recursive rules for linear and exponential functions. You can use
the pattern in the fi rst differences to write recursive rules for quadratic functions.
Writing a Recursive Rule
Write a recursive rule for the quadratic function f in Example 3.
SOLUTION
An expression for the nth term of the sequence of fi rst differences is −12 + 8(n − 1),
or 8n − 20. Notice that f(n) − f(n − 1) = 8n − 20.
So, a recursive rule for the quadratic function is f(0) = 12,
f(n) = f(n − 1) + 8n − 20.
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5. Tell whether the table of values represents a
linear, an exponential, or a quadratic
function. Then write the function.
6. Write a recursive rule for the function
represented by the table of values.
x 0 1 2 3 4
f(x) 12 0 −4 0 12
STUDY TIPTo check your function in Example 3, substitute the other points from the table to verify that they satisfy the function.
x −1 0 1 2 3
y 16 8 4 2 1
x 0 1 2 3 4
f(x) 12 0 −4 0 12
+ 1
− 12
+ 1
− 4
+ 8
+ 1 + 1
+ 4
+ 8
+ 12
+ 8
first differences
second differences
x 0 1 2 3 4
f(x) −8 −1 10 25 44
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Section 3.7 Comparing Linear, Exponential, and Quadratic Functions 173
Comparing Functions Using Average Rates of ChangeFor nonlinear functions, the rate of change is not constant. You can compare two
nonlinear functions over the same interval using their average rates of change. Recall
that the average rate of change of a function y = f (x) between x = a and x = b is the
slope of the line through (a, f (a)) and (b, f (b)).
average rate of change = change in y
— change in x
= f (b) − f (a) —
b − a
Exponential Function Quadratic Function
x
y
ab − a
f(a)
f(b)
f(b) − f(a)
b x
y
b − af(a)
f(b) − f(a)
f(b)
a b
Using and Interpreting Average Rates of Change
The table and graph show the populations of two species of fi sh introduced into a lake.
Compare the populations by calculating and interpreting the average rates of change
from Year 0 to Year 4.
Species A
Year, x Population, y
0 100
2 264
4 428
6 592
8 756
10 920
Species B
Pop
ula
tio
n0
200
400
600
Year6420 x
y
SOLUTION
Calculate the average rates of change by using the points whose x-coordinates
are 0 and 4.
Species A: Use (0, 100) and (4, 428).
average rate of change = f (b) − f (a)
— b − a
= 428 − 100
— 4 − 0
= 82
Species B: Use the graph to estimate the points when x = 0 and x = 4. Use (0, 100)
and (4, 300).
average rate of change = f (b) − f (a)
— b − a
≈ 300 − 100 —
4 − 0 = 50
From Year 0 to Year 4, the population of Species A increases at an average rate
of 82 fi sh per year, and the population of Species B increases at an average rate of
about 50 fi sh per year. So, the population of Species A is growing faster.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
7. Compare the populations by calculating and interpreting the average rates of
change from Year 4 to Year 8.
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174 Chapter 3 Graphing Quadratic Functions
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
8. From 1950 to 2000, which town’s population had a greater average rate of change?
Comparing Different Function Types
Let x represent the number of years since 1900. The function C(x) = 12x2 + 12x + 100
represents the population of Cedar Ridge. In 1900, Pine Valley had a population of
50 people. Pine Valley’s population increased by 9% each year.
a. From 1900 to 1950, which town’s population had a greater average rate of change?
b. Which town will eventually have a greater population? Explain.
SOLUTION
a. Write the function that represents the population of Cedar Ridge and a function to
model the population of Pine Valley.
Cedar Ridge: C(x) = 12x2 + 12x + 100 Quadratic function
Pine Valley: P(x) = 50(1.09)x Exponential function
Find the average rate of change from 1900 to 1950 for the population of each town.
Cedar Ridge Pine Valley
C(50) − C(0) ——
50 − 0 = 612
P(50) − P(0) ——
50 − 0 ≈ 73
From 1900 to 1950, the average rate of change of Cedar Ridge’s population
was greater.
b. Because Pine Valley’s population is given by an increasing exponential function and
Cedar Ridge’s population is given by an increasing quadratic function, Pine Valley
will eventually have a greater population. Using a graphing calculator, you can see
that Pine Valley’s population caught up to and exceeded Cedar Ridge’s population
between x = 87 and x = 88, which corresponds to 1987.
1000
0
200,000
Y=92627.727IntersectionX=87.311696
Pine Valley
Cedar Ridge
X Y1
X=87
90173909539174092534933359414394958
Y291972921829239292603928149302593236
87.187
87.287.387.487.587.6
PineValley
CedarRidge
Core Core ConceptConceptComparing Functions Using Average Rates of Change• As a and b increase, the average rate of change between x = a and x = b of an
increasing exponential function y = f(x) will eventually exceed the average rate
of change between x = a and x = b of an increasing quadratic function
y = g(x) or an increasing linear function y = h(x). So, as x increases, f(x) will
eventually exceed g(x) or h(x).
• As a and b increase, the average rate of change between x = a and x = b of an
increasing quadratic function y = g(x) will eventually exceed the average rate
of change between x = a and x = b of an increasing linear function y = h(x).
So, as x increases, g(x) will eventually exceed h(x).
STUDY TIPYou can explore these concepts using a graphing calculator.
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Section 3.7 Comparing Linear, Exponential, and Quadratic Functions 175
Exercises3.7 Dynamic Solutions available at BigIdeasMath.com
In Exercises 5–8, tell whether the points appear to represent a linear, an exponential, or a quadratic function.
5.
4
2
6
x−4 −2
y 6.
x
y
8
12
4
2 4
7.
4
6
2
−2
x2−2
y 8.
2
−4
x2−2
y
In Exercises 9–14, plot the points. Tell whether the points appear to represent a linear, an exponential, or a quadratic function. (See Example 1.)
9. (−2, −1), (−1, 0), (1, 2), (2, 3), (0, 1)
10. ( 0, 1 —
4 ) , (1, 1), (2, 4), (3, 16), ( −1,
1 —
16 )
11. (0, −3), (1, 0), (2, 9), (−2, 9), (−1, 0)
12. (−1, −3), (−3, 5), (0, −1), (1, 5), (2, 15)
13. (−4, −4), (−2, −3.4), (0, −3), (2, −2.6), (4, −2)
14. (0, 8), (−4, 0.25), (−3, 0.4), (−2, 1), (−1, 3)
In Exercises 15–18, tell whether the table of values represents a linear, an exponential, or a quadratic function. (See Example 2.)
15. x −2 −1 0 1 2
y 0 0.5 1 1.5 2
16. x −1 0 1 2 3
y 0.2 1 5 25 125
17. x 2 3 4 5 6
y 2 6 18 54 162
18. x −3 −2 −1 0 1
y 2 4.5 8 12.5 18
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. WRITING Name three types of functions that you can use to model data. Describe the equation and
graph of each type of function.
2. WRITING How can you decide whether to use a linear, an exponential, or a quadratic function
to model a data set?
3. VOCABULARY Describe how to fi nd the average rate of change of a function y = f (x) between
x = a and x = b.
4. WHICH ONE DOESN’T BELONG? Which graph does not belong with the other three? Explain
your reasoning.
2
4
x2−2
y
f
1
−2
−4
x2−2
y
m
2
x−4 −2
yn
−4
x2−2
y
g
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
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176 Chapter 3 Graphing Quadratic Functions
19. MODELING WITH MATHEMATICS A student takes
a subway to a public library. The table shows the
distances d (in miles) the student travels in t minutes.
Let the time t represent the independent variable.
Tell whether the data can be modeled by a linear,
an exponential, or a quadratic function. Explain.
Time, t 0.5 1 3 5
Distance, d 0.335 0.67 2.01 3.35
20. MODELING WITH MATHEMATICS A store sells
custom circular rugs. The table
shows the costs c (in dollars)
of rugs that have diameters
of d feet. Let the diameter d
represent the independent
variable. Tell whether the
data can be modeled by a
linear, an exponential, or
a quadratic function. Explain.
Diameter, d 3 4 5 6
Cost, c 63.90 113.60 177.50 255.60
In Exercises 21–26, tell whether the data represent a linear, an exponential, or a quadratic function. Then write the function. (See Example 3.)
21. (−2, 8), (−1, 0), (0, −4), (1, −4), (2, 0), (3, 8)
22. (−3, 8), (−2, 4), (−1, 2), (0, 1), (1, 0.5)
23. x −2 −1 0 1 2
y 4 1 −2 −5 −8
24. x −1 0 1 2 3
y 2.5 5 10 20 40
25. 26.
−4
2
−8
x2−2−4
y
−12
2
4
x2−2−4
y
−2
−4
In Exercises 27 and 28, write a recursive rule for the function represented by the table of values. (See Example 4.)
27. x 0 1 2 3 4
f(x) 2 1 2 5 10
28. x 0 1 2 3 4
f (x) 3 6 12 24 48
29. ERROR ANALYSIS Describe and correct the error in
determining whether the table represents a linear, an
exponential, or a quadratic function.
+ 1
× 3
+ 1
× 3
+ 1 + 1
× 3 × 3
x 1 2 3 4 5
y 3 9 27 81 243
Consecutive y-values change by a
constant amount. So, the table
represents a linear function.
✗
30. ERROR ANALYSIS Describe and correct the error in
writing the function represented by the table.
x −3 −2 −1 0 1
y 4 0 −2 −2 0
+ 1
− 4
+ 1
− 2
+ 2
+ 1 + 1
+ 0
+ 2
+ 2
+ 2
first differences
second differences
The table represents a quadratic function.
f (x) = a(x − 2)(x − 1)
4 = a(−3 − 2)(−3 − 1)
1 —
5 = a
f (x) = 1
— 5
(x − 2)(x − 1)
= 1
— 5
x2 − 3 — 5
x + 2
— 5
So, the function is f (x) = 1 — 5
x2 − 3 — 5
x + 2 — 5
.
✗
31. REASONING The table shows the numbers of people
attending the fi rst fi ve football games at a high school.
Game, g 1 2 3 4 5
People, p 252 325 270 249 310
a. Plot the points. Let the game g represent the
independent variable.
b. Can a linear, an exponential, or a quadratic
function represent this situation? Explain.
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Section 3.7 Comparing Linear, Exponential, and Quadratic Functions 177
32. MODELING WITH MATHEMATICS The table shows
the breathing rates y (in liters of air per minute) of
a cyclist traveling at different speeds x (in miles
per hour).
Speed, x 20 21 22 23 24
Breathing rate, y
51.4 57.1 63.3 70.3 78.0
a. Plot the points. Let the speed
x represent the independent
variable. Then determine the
type of function that best
represents this situation.
b. Write a function that
models the data.
c. Find the breathing rate of a
cyclist traveling 18 miles per hour. Round your
answer to the nearest tenth.
33. ANALYZING RATES OF CHANGE The function
f (t) = −16t2 + 48t + 3 represents the height (in feet)
of a volleyball t seconds after it is hit into the air.
a. Copy and complete the table.
t 0 0.5 1 1.5 2 2.5 3
f (t)
b. Plot the ordered pairs and draw a smooth curve
through the points.
c. Describe where the function is increasing
and decreasing.
d. Find the average rate of change for each
0.5-second interval in the table. What do you
notice about the average rates of change when
the function is increasing? decreasing?
34. ANALYZING RELATIONSHIPS The population of
Species A in 2010 was 100. The population of
Species A increased by 6% every year. Let x represent
the number of years since 2010. The graph shows the
population of Species B. Compare the populations of
the species by calculating
and interpreting the
average rates of change
from 2010 to 2016. (See Example 5.)
35. ANALYZING RELATIONSHIPS Three charities are
holding telethons. Charity A begins with one pledge,
and the number of pledges triples each hour. The table
shows the numbers of pledges received by Charity B.
The graph shows the numbers of pledges received by
Charity C.
Time (hours),
t
Number of pledges,
y
0 12
1 24
2 36
3 48
4 60
5 72
6 84
a. What type of function represents the numbers of
pledges received by Charity A? B? C?
b. Find the average rates of change of each function
for each 1-hour interval from t = 0 to t = 6.
c. For which function does the average rate of
change increase most quickly? What does this tell
you about the numbers of pledges received by the
three charities?
36. COMPARING FUNCTIONS Let x represent the number
of years since 1900. The function
H(x) = 10x2 + 10x + 500
represents the population of Oak Hill. In 1900, Poplar
Grove had a population of 200 people. Poplar Grove’s
population increased by 8% each year.
(See Example 6.)
a. From 1900 to 1950, which town’s population had
a greater average rate of change?
b. Which town will eventually have a greater
population? Explain.
37. COMPARING FUNCTIONS Let x represent the number
of years since 2000. The function
R(x) = 0.01x2 + 0.22x + 1.08
represents the revenue (in millions of dollars) of
Company A. In 2000, Company B had a revenue of
$2.12 million. Company B’s revenue increased by
$0.32 million each year.
a. From 2000 to 2015, which company’s revenue had
a greater average rate of change?
b. Which company will eventually have a greater
revenue? Explain.
Species B
Pop
ula
tio
n
0
50
100
150
Years since 2010840 x
y
t
y
80
40
0
160
120
420 6
(2, 16)(3, 36)
(0, 0)(1, 4) (4, 64)
Time (hours)
Nu
mb
er o
f p
led
ges
Charity C
(6, 144)
(5, 100)
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178 Chapter 3 Graphing Quadratic Functions
38. HOW DO YOU SEE IT? Match each graph with its
function. Explain your reasoning.
a. y
x
b. y
x
c. y
x
d. y
x
A. y = 2x2 − 4 B. y = 2x − 1
C. y = 2x − 1 D. y = −2x2 + 4
39. REASONING Explain why the average rate of change
of a linear function is constant and the average rate
of change of a quadratic or exponential function is
not constant.
40. CRITICAL THINKING In the ordered pairs below, the
y-values are given in terms of n. Tell whether the
ordered pairs represent a linear, an exponential, or a
quadratic function. Explain.
(1, 3n − 1), (2, 10n + 2), (3, 26n),
(4, 51n − 7), (5, 85n − 19)
41. USING STRUCTURE Write a function that has constant
second differences of 3.
42. CRITICAL THINKING Is the graph of a set of points
enough to determine whether the points represent a
linear, an exponential, or a quadratic function? Justify
your answer.
43. MAKING AN ARGUMENT Function p is an
exponential function and
function q is a quadratic
function. Your friend says
that after about x = 3,
function q will always
have a greater y-value than
function p. Is your friend
correct? Explain.
44. THOUGHT PROVOKING Find three different patterns
in the fi gure. Determine whether each pattern
represents a linear, an exponential, or a quadratic
function. Write a model for each pattern.
n = 1 n = 2 n = 3 n = 4
45. USING TOOLS The table shows the amount a
(in billions of dollars) United States residents spent
on pets or pet-related products and services each
year for a 5-year period. Let the year x represent
the independent variable. Using technology, fi nd a
function that models the data. How did you choose
the model? Predict how much residents will spend on
pets or pet-related products and services in Year 7.
Year, x 1 2 3 4 5
Amount, a 53.1 56.9 61.8 65.7 67.1
x
y
4
8
12
2 4 6 8
pq
Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyEvaluate the expression. (Section 1.5)
46. √—
121 47. 3 √—
125
48. 3 √—
512 49. 5 √—
243
Find the product. (Section 2.3)
50. (x + 6)(x − 6) 51. (2y + 5)(2y − 5)
52. (4c − 3d)(4c + 3d) 53. (−3s + 8t)(−3s − 8t)
Reviewing what you learned in previous grades and lessons
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179
3.4–3.7 What Did You Learn?
Core VocabularyCore Vocabularyeven function, p. 146odd function, p. 146
vertex form (of a quadratic
function), p. 148intercept form, p. 154focus, p. 162directrix, p. 162
Core ConceptsCore ConceptsSection 3.4Even and Odd Functions, p. 146Graphing f (x) = a(x − h)2, p. 147 Graphing f (x) = a(x − h)2 + k, p. 148
Writing Quadratic Functions of the Form
f (x) = a(x − h)2 + k, p. 149
Section 3.5Graphing f (x) = a(x − p)(x − q), p. 154Factors and Zeros, p. 156
Using Characteristics to Graph and Write Quadratic
Functions, p. 156
Section 3.6Standard Equations of a Parabola with Vertex at the
Origin, p. 163Standard Equations of a Parabola with Vertex at (h, k),
p. 164
Section 3.7Linear, Exponential, and Quadratic Functions, p. 170Differences and Ratios of Functions, p. 171
Writing Functions to Model Data, p. 172Comparing Functions Using Average Rates of Change,
p. 173
Mathematical PracticesMathematical Practices1. How can you use technology to confi rm your answer in Exercise 64 on page 152?
2. How did you use the structure of the equation in Exercise 65 on page 159 to solve the problem?
3. Describe why your answer makes sense considering the context of the data in Exercise 20 on page 176.
Performance Task:
Solar EnergyThere are many factors that infl uence the design of a product, such as aesthetics, available space, and special properties. What are the special properties of a parabola that contribute to its use? How do parabolas help create power from solar energy?
To explore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.
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180 Chapter 3 Graphing Quadratic Functions
33 Chapter Review
Graphing f (x) = ax2 (pp. 123–128)3.1
Graph g(x) = −4x2. Compare the graph to the graph of f (x) = x2.
Step 1 Make a table of values.
x −2 −1 0 1 2
g(x) −16 −4 0 −4 −16
Step 2 Plot the ordered pairs.
Step 3 Draw a smooth curve through the points.
The graphs have the same vertex, (0, 0), and the same axis of symmetry, x = 0, but the graph
of g opens down and is narrower than the graph of f. So, the graph of g is a vertical stretch by
a factor of 4 and a refl ection in the x-axis of the graph of f.
Graph the function. Compare the graph to the graph of f (x) = x2.
1. p(x) = 7x2 2. q(x) = 1 —
2 x2 3. g(x) = − 3 —
4 x2 4. h(x) = −6x2
5. Identify characteristics of the
quadratic function and its graph.
Graphing f (x) = ax2 + c (pp. 129–134)3.2
Graph g(x) = 2x2 + 3. Compare the graph to the graph of f (x) = x2.
Step 1 Make a table of values.
x −2 −1 0 1 2
g(x) 11 5 3 5 11
Step 2 Plot the ordered pairs.
Step 3 Draw a smooth curve through the points.
Both graphs open up and have the same axis of symmetry, x = 0.
The graph of g is narrower, and its vertex, (0, 3), is above the vertex of the graph of f, (0, 0). So, the
graph of g is a vertical stretch by a factor of 2 and a vertical translation 3 units up of the graph of f.
Graph the function. Compare the graph to the graph of f (x) = x2.
6. g(x) = x2 + 5 7. h(x) = −x2 − 4 8. m(x) = −2x2 + 6 9. n(x) = 1 —
3 x2 − 5
−8
−16
8
x42−2−4
y
f(x) = x2
g(x) = −4x2
−1−3 1 3 x
y
−2
2
6
10
14
x42−2−4
y14
y
g(x) = 2x2 + 3
f(x) = x2
Dynamic Solutions available at BigIdeasMath.com
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Chapter 3 Chapter Review 181
Graphing f (x) = ax2 + bx + c (pp. 135–142)3.3
Graph f (x) = 4x2 + 8x − 1. Describe the domain and range.
Step 1 Find and graph the axis of symmetry: x = − b —
2a = −
8 —
2(4) = −1.
Step 2 Find and plot the vertex. The axis of symmetry is x = −1. So, the x-coordinate of the vertex
is −1. The y-coordinate of the vertex is f (−1) = 4(−1)2 + 8(−1) − 1 = −5. So, the vertex
is (−1, −5).
Step 3 Use the y-intercept to fi nd two more points on the graph.
Because c = −1, the y-intercept is −1. So, (0, −1) lies on
the graph. Because the axis of symmetry is x = −1, the
point (−2, −1) also lies on the graph.
Step 4 Draw a smooth curve through the points.
The domain is all real numbers. The range is y ≥ −5.
Graph the function. Describe the domain and range.
10. y = x2 − 2x + 7 11. f (x) = −3x2 + 3x − 4 12. y = 1 —
2 x2 − 6x + 10
13. The function f (t) = −16t2 + 88t + 12 represents the height (in feet) of a pumpkin t seconds
after it is launched from a catapult. When does the pumpkin reach its maximum height? What is
the maximum height of the pumpkin?
Graphing f (x) = a(x − h)2 + k (pp. 145–152)3.4
Determine whether f (x) = 2x2 + 4 is even, odd, or neither.
f (x) = 2x2 + 4 Write the original function.
f (−x) = 2(−x)2 + 4 Substitute −x for x.
= 2x2 + 4 Simplify.
= f (x) Substitute f (x) for 2x2 + 4.
Because f (−x) = f (x), the function is even.
Determine whether the function is even, odd, or neither.
14. w(x) = 5x 15. r(x) = −8x 16. h(x) = 3x2 − 2x
Graph the function. Compare the graph to the graph of f (x) = x2.
17. h(x) = 2(x − 4)2 18. g(x) = 1 —
2 (x − 1)2 + 1 19. q(x) = −(x + 4)2 + 7
20. Consider the function g(x) = −3(x + 2)2 − 4. Graph h(x) = g(x − 1).
21. Write a quadratic function whose graph has a vertex of (3, 2) and passes through the point (4, 7).
2
x1−3−5−7
y
f(x) = 4x2 + 8x − 1
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182 Chapter 3 Graphing Quadratic Functions
Graphing f(x) = a(x − p)(x − q) (pp. 153–160)3.5
a. Graph f (x) = −(x + 4)(x − 2). Describe the domain and range.
Step 1 Identify the x-intercepts. Because the x-intercepts are p = −4 and q = 2, plot (−4, 0)
and (2, 0).
Step 2 Find and graph the axis of symmetry.
x = p + q
— 2 =
−4 + 2 —
2 = −1
Step 3 Find and plot the vertex.
The x-coordinate of the vertex is −1.
To fi nd the y-coordinate of the vertex,
substitute −1 for x and simplify.
f (−1) = −(−1 + 4)(−1 − 2) = 9
So, the vertex is (−1, 9).
Step 4 Draw a parabola through the vertex and
the points where the x-intercepts occur.
The domain is all real numbers. The range is y ≤ 9.
b. Use zeros to graph h(x) = x2 − 7x + 6.
The function is in standard form. The parabola opens up (a > 0), and the y-intercept is 6.
So, plot (0, 6).
The polynomial that defi nes the function is factorable. So, write the function in intercept form and
identify the zeros.
h(x) = x2 − 7x + 6 Write the function.
= (x − 6)(x − 1) Factor the trinomial.
The zeros of the function are 1 and 6. So, plot (1, 0) and (6, 0). Draw a parabola through the points.
2 4 x
y
−4
−8
4
h(x) = x2 − 7x + 6
Graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts. Describe the domain and range of the function.
22. y = (x − 4)(x + 2) 23. f (x) = −3(x + 3)(x + 1) 24. y = x2 − 8x + 15
Use zeros to graph the function.
25. y = −2x2 + 6x + 8 26. f (x) = x2 + x − 2 27. f (x) = 2x2 − 18
28. Write a quadratic function in standard form whose graph passes through (4, 0) and (6, 0).
x
y
4
6
8
10
2
4−2
(2, 0)(−4, 0)
(−1, 9) f(x) = −(x + 4)(x − 2)
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Chapter 3 Chapter Review 183
Focus of a Parabola (pp. 161–168)3.6
a. Identify the focus, directrix, and axis of symmetry of 8x = y2. Graph the equation.
Step 1 Rewrite the equation in standard form.
8x = y2 Write the original equation.
x = 1 —
8 y2 Divide each side by 8.
Step 2 Identify the focus, directrix, and axis of symmetry. The equation has the
form x = 1 —
4p y2, where p = 2. The focus is (p, 0), or (2, 0). The directrix is
x = −p, or x = −2. Because y is squared, the axis of symmetry is the x-axis.
Step 3 Use a table of values to graph the equation. Notice that it is easier to substitute
y-values and solve for x.
y 0 ±2 ±4 ±6
x 0 0.5 2 4.5
x
y8
4
84(2, 0)
−4
x = −2
b. Write an equation of the parabola shown.
Because the vertex is not at the origin and the axis
vertex
focus
x
y
6
8
2
4 62−2
of symmetry is vertical, the equation has the form
y = 1 —
4p (x − h)2 + k. The vertex (h, k) is (2, 3) and
the focus (h, k + p) is (2, 4), so h = 2, k = 3, and p = 1.
Substitute these values to write an equation of the parabola.
y = 1 —
4(1) (x − 2)2 + 3 =
1 —
4 (x − 2)2 + 3
An equation of the parabola is y = 1 —
4 (x − 2)2 + 3.
29. You can make a solar hot-dog cooker by shaping foil-lined
cardboard into a parabolic trough and passing a wire through
the focus of each end piece. For the trough shown, how far
from the bottom should the wire be placed?
30. Graph the equation 36y = x2. Identify the focus,
directrix, and axis of symmetry.
Write an equation of the parabola with the given characteristics.
31. vertex: (0, 0) 32. focus: (2, 2)
directrix: x = 2 vertex: (2, 6)
acteristics.
4 in.
12 in.
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184 Chapter 3 Graphing Quadratic Functions
Comparing Linear, Exponential, and Quadratic Functions (pp. 169−178)3.7
Tell whether the data represent a linear, an exponential, or a quadratic function.
a. (−4, 1), (−3, −2), (−2, −3)
(−1, −2), (0, 1)
−3
2
x−4 −2
y
b. (−2, 16), (−1, 8), (0, 4)
(1, 2), (2, 1)
x
y
16
8
2−2
The points appear to represent
a quadratic function.
The points appear to represent
an exponential function.
c. x −1 0 1 2 3
y 15 8 1 −6 −13
+ 1
− 7 − 7 − 7 − 7
+ 1 + 1 + 1
x −1 0 1 2 3
y 15 8 1 −6 −13
d. x 0 1 2 3 4
y 1 5 6 4 −1
The fi rst differences are constant. So,
the table represents a linear function.
The second differences are constant.
So, the table represents a quadratic
function.
33. Tell whether the table of values represents a linear, an exponential, or a quadratic function.
Then write the function.
x −1 0 1 2 3
y 512 128 32 8 2
34. Write a recursive rule for the function represented by the table of values.
x 0 1 2 3 4
f (x) 5 6 11 20 33
35. The balance y (in dollars) of your savings account after t years is represented by y = 200(1.1)t. The beginning balance of your friend’s account is $250, and the balance increases by
$20 each year. Compare the account balances by calculating and interpreting the average rates
of change from t = 2 to t = 7.
+ 1
+ 4
+ 1
+ 1
− 3
+ 1 + 1
− 2
− 3
− 5
− 3
x 0 1 2 3 4
y 1 5 6 4 −1
first differences
second differences
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Chapter 3 Chapter Test 185
Chapter Test33Graph the function. Compare the graph to the graph of f (x) = x2.
1. h(x) = 2x2 − 3 2. g(x) = − 1 — 2 x2 3. p(x) =
1 —
2 (x + 1)2 − 1
4. Consider the graph of the function f.
a. Find the domain, range, and zeros of the function.
b. Write the function f in standard form.
c. Compare the graph of f to the graph of g(x) = x2.
d. Graph h(x) = f (x − 6).
Use zeros to graph the function. Describe the domain and range of the function.
5. f (x) = 2x2 − 8x + 8 6. y = −(x + 5)(x − 1) 7. h(x) = 16x2 − 4
8. Identify an focus, directrix, and axis of symmetry of x = 2y2. Graph the equation.
Write an equation of the parabola. Justify your answer.
9.
x
y
4
−4−8
focus−4
vertex
10.
x
y
2
2 4 6
−2
11.
x
y
1
3
−3 −1−5 1 3
directrix −3
vertex
Tell whether the table of values represents a linear, an exponential, or a quadratic function. Explain your reasoning. Then write the function.
12. x −1 0 1 2 3
y 4 8 16 32 64
13. x −2 −1 0 1 2
y −8 −2 0 −2 −8
14. You are playing tennis with a friend. The path of the tennis ball after you return a serve
can be modeled by the function y = −0.005x2 + 0.17x + 3, where x is the horizontal
distance (in feet) from where you hit the ball and y is the height (in feet) of the ball.
a. What is the maximum height of the tennis ball?
b. You are standing 30 feet from the net, which is 3 feet high. Will the ball clear the net?
Explain your reasoning.
15. Find values of a, b, and c so that the function f (x) = ax2 + bx + c is (a) even, (b) odd, and
(c) neither even nor odd.
16. Consider the function f (x) = x2 + 4. Find the average rate of change from x = 0 to x = 1,
from x = 1 to x = 2, and from x = 2 to x = 3. What do you notice about the average rates
of change when the function is increasing?
8
4
2 4 6 8 x
y
(3, 0) (7, 0)
(5, 8)
f
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186 Chapter 3 Graphing Quadratic Functions
33 Cumulative Assessment
1. Which function is represented by the graph?
x
y
−2
−4
−2 2
○A y = 1 —
2 x2 ○B y = 2x2
○C y = − 1 —
2 x2 ○D y = −2x2
2. A linear function f has the following values at x = 2 and x = 5.
f (2) = 6, f (5) = 3
What is the value of the inverse of f at x = 4?
3. The function f (t) = −16t2 + v0t + s0 represents the height (in feet) of a ball t seconds
after it is thrown from an initial height s0 (in feet) with an initial vertical velocity v0
(in feet per second). The ball reaches its maximum height after 7 —
8 second when it is
thrown with an initial vertical velocity of ______ feet per second.
4. Order the following parabolas from widest to narrowest.
A. focus: (0, −3); directrix: y = 3 B. y = 1 —
16 x2 + 4
C. x = 1 —
8 y2 D. y =
1 —
4 (x − 2)2 + 3
5. Which polynomial represents the area (in square feet) of the shaded region of the fi gure?
x ft
x ft
a ft
a ft
○A a2 − x2 ○B x2 − a2
○C x2 − 2ax + a2 ○D x2 + 2ax + a2
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Chapter 3 Cumulative Assessment 187
6. Consider the functions represented by the tables.
x 0 1 2 3
p(x) −4 −16 −28 −40
x 1 2 3 4
r(x) 0 15 40 75
x 1 2 3 4
s(x) 72 36 18 9
x 1 3 5 7
t(x) 3 −5 −21 −45
a. Classify each function as linear, exponential, or quadratic.
b. Order the functions from least to greatest according to the average rates of
change between x = 1 and x = 3.
7. Which expressions are equivalent to 14x − 15 + 8x2?
(4x + 3)(2x − 5)
(4x – 3)(2x − 5)(4x + 3)(2x + 5)
(4x – 3)(2x + 5)
8x2 − 14x − 15 8x2 + 14x − 15
8. Complete each function using the symbols + or − , so that the graph of the quadratic
function satisfi es the given conditions.
a. f (x) = 5(x 3)2 4; vertex: (−3, 4)
b. g(x) = −(x 2)(x 8); x-intercepts: −8 and 2
c. h(x) = 3x2 6; range: y ≥ −6
d. j(x) = 4(x 1)(x 1); range: y ≤ 4
9. Which expressions are equivalent to (b−5)−4?
b−20 b−9(b−4)−5b−6b−14
b−5b−4 b20(b10)2
(b−2)−7
b12b8(b−12)3
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